On f -domination: polyhedral and algorithmic results

Given an undirected simple graph G with node set V and edge set E , let f v , for each node v ∈ V , denote a nonnegative integer value that is lower than or equal to the degree of v in G . An f -dominating set in G is a node subset D such that for each node v ∈ V \ D at least f v of its neighbors belong to D . In this paper, we study the polyhedral structure of the polytope deﬁned as the convex hull of all the incidence vectors of f -dominating sets in G and give a complete description for the case of trees. We prove that the corresponding separation problem can be solved in polynomial time. In addition, we present a linear-time algorithm to solve the weighted version of the problem on trees: Given a cost c v ∈ R for each node v ∈ V , ﬁnd an f -dominating set D in G whose cost, given by (cid:2) v ∈ D c v , is a minimum.

N G [S] = N G (S) ∪ S. In case S consists of a single node v ∈ V we may write N G (v) (resp. N G [v]) instead of N G ({v}) (resp. N G [{v}]). Given some node v ∈ V , let d G v denote the degree of node v in G, i.e., d G v = |N G (v)|. When G is clear from the context we shall more simply write d v , and we may also omit G in the subscript of the notation used for neighborhoods as well. Let c ∈ R n + denote a vector of nonnegative node weights. Let F G stand for the following set of vectors indexed on the nodes of G: has at least f v neighbors in D, i.e., |N (v) ∩ D| ≥ f v . If, in addition, all the nodes in D (and thus all the nodes in V ) also satisfy this inequality, then D is called a total f -dominating set. We will call f v the domination requirement for node v.
The f -domination concept seems to appear first in the papers by Hedetniemi et al. (1985) and Stracke and Volkmann (1993). The concepts of f -domination and f -total domination generalize those of domination and total domination, respectively. The two latter correspond to the special case when f v = 1, for all v ∈ V . (For further results related to the domination or total domination concepts and variants the reader may consult e.g. Haynes et al. (1998a), Haynes et al. (1998b), Henning and Yeo (2013), Bermudo et al. (2018) and Zhou (1996), and the references therein.) The authors previously investigated the polyhedral structures of polytopes related to the set of all total f -dominating sets in a graph (Dell'Amico et al. 2017). Here, we proceed to such investigations with respect to the f -dominating sets. Despite the apparent similarity of the two concepts, the polyhedral structures of the corresponding polytopes radically differ. And this also holds if we compare the polytopes corresponding, on the one hand, to the classical domination, and on the other, to the generalized f -domination concept we consider here.
In addition to a purely theoretical interest of these structures, the obtained results are relevant w.r.t. the problem which consists in determining a minimum weight fdominating set in a graph G, where we define the weight of any node subset set S ⊆ V as the quantity v∈S c v . This problem, denoted by [M W f ] is described hereafter together with an integer linear programming formulation. [MW f ] Minimum weight f -dominating set problem Find a minimum weight f -dominating set of G, i.e., find a node subset S ⊆ V such that S is an f -dominating set and the weight v∈S c v of S is minimum.
This problem may be formulated as the following integer program.
With no loss of generality we may assume that all the weights (or costs) (c v ) v∈V are positive. If this is not the case, the nodes in V with a nonpositive weight can be fixed in any optimal solution. Let 1 stand for the n-dimensional all ones vector. The particular case when f = c = 1 corresponds to the classical minimum (cardinality) dominating set problem. Given a node subset S ⊆ V , let χ S ∈ {0, 1} n denote its incidence vector: G denote the f -dominating set polytope, i.e., the convex hull in R n of all the incidence vectors of the f -dominating sets in G. Then, problem [M W f ] can be reformulated as the linear program: min{c t x : x ∈ D f G }. Also, note that for any pair Optimization problems involving dominating sets arise in many classical applications (see, e.g., Haynes et al. 1998a, b) that received renewed interest in the last twenty years. We refer, among others, to problems in telecommunication networks, such as clustering, backbone formation and intrusion detection in wireless ad-hoc and sensor networks (Wu and Li 1999;Chen and Liestman 2002;Subhadrabandhu et al. 2004), the gateway placement in wireless mesh networks (Aoun et al. 2006), and the deployment of wavelength division multiplexing in optical networks (Houmaidi et al. 2003). Data analysis problems such as information retrieval for multi document summarization (Shen and Li 2010) and query selection on web databases (Wu et al. 2006), are another domain of application where dominating sets are largely used.
More recently the minimum weighted dominating set problem, which is the natural extension of the minimum dominating set problem obtained by adding weights to the nodes, attracted several researchers and practitioners (see, e.g., Zou et al. 2011;Potluri and Singh 2013;Bouamama and Blum 2016).
The problem we consider in this paper corresponds to a further generalization. Given a wireless sensor network, represented by graph G = (V , E), we wish to determine a minimum weight subset of nodes S to be upgraded as cluster heads, such that each node v ∈ V \S has at least f v neighbors in S, where f v is some given nonnegative integer. This consists in identifying a minimum weight fault-tolerant dominating set, i.e., a minimum weight node subset S such that if some set F of q nodes fail (for example, due to some environmental conditions), the set of nodes v ∈ V \F for which f v ≥ q + 1 or v ∈ S\F are still dominated by S\F (i.e., these nodes either belong to S\F or have at least one neighbor in S\F). The case when all the nodes have unit weights and f v = k, v ∈ V , for some given positive integer k is presented, e.g., in (Couture et al. 2006;Hwang and Chang 1991;Foerster 2013), see also the references therein.

Related work
The minimum cardinality dominating set problem can be solved in linear time if the graph G is series-parallel (Kikuno et al. 1983) or a cactus (Hedetniemi et al. 1986). It is known to be N P-hard in planar graphs with maximum node degree 3, regular planar graphs with nodes of degree 4 (Garey and Johnson 1979), chordal graphs (Booth 1980) and undirected path graphs (Booth and Johnson 1982). Also, for this problem, there exists a polynomial time H Δ+1 − 1 2 -approximation algorithm, where Δ stands for the maximum degree in G, and, for some given positive integer q, H q = q i=1 1 i denotes the qth harmonic number (Chlebík and Chlebíková 2008;Duh and Fürer 1997). But it is N P-hard to approximate within a factor of ln(n)−c ln ln Δ for general graphs with Δ ≥ Δ 0 , where c > 0 and Δ 0 ≥ 3 are absolute constants (Chlebík and Chlebíková 2008)). For the minimum cardinality f -dominating set in block graphs, Hwang and Chang 1991 presented a linear time algorithm. In this paper we extend their results to the weighted case for trees. For the case when f v = k, v ∈ V , for some positive integer k, Foerster (2013) showed that the minimum cardinality f -dominating set problem is N P-hard to approximate with a factor better than 0.2267/k ln(n/k), and he also provided a greedy-type algorithm with an approximation ratio of ln(Δ + k) + 1 <ln(Δ)+1.7. For the case of unit disk graphs and Couture et al. 2006 introduced an incremental algorithm having a constant deterministic performance ratio of six (see also the references therein). We now report some works dealing with the minimum weight dominating set problem, i.e., [M W 1 ] with linear programming based approaches. Given the afore mentioned complexity results, no complete description of the polytope D 1 G is presently known for general graphs. For the case of strongly chordal graphs, the linear relaxation of (I P1) provides such a description, i.e., we have D 1 u∈N [v] x u ≥ 1, v ∈ V }, and this implies the polynomial time solvability of [M W 1 ] for this graph family (Farber 1984). A complete description of D 1 G for the case of domishold graphs appears in (Mahjoub 1983). A graph is said to be domishold (Benzaken and Hammer 1978) if there exist real positive weights associated to the nodes so that a node subset is a dominating set if and only if the sum of the corresponding weights exceeds some threshold value. A complete formulation for cycles firstly appears in (Bouchakour et al. 2008). This work was extended in (Bianchi et al. 2010), leading to complete description for webs of the form W k s(2k+1)+r with s = 2, 3 and 0 ≤ r ≤ s − 1. Also recently, an exact extended formulation for cacti graphs was introduced in (Baïou and Barahona 2014), together with a polynomial-time algorithm to solve [M W 1 ] for cacti. Given two graphs is a clique with cardinality k ≥ 1, the graph G = (V 1 ∪ V 2 , E 1 ∪ E 2 ) is called the k-sum of G 1 and G 2 . For the particular case when k = 1 and assuming complete formulations of the polytopes D 1 G i , i = 1, 2, are known, a procedure for determining a complete formulation of D 1 G is presented by Bouchakour and Mahjoub (1997). It follows from the latter that, in addition to the graph classes we mentioned just before (i.e., those for which a complete formulation is known), complete formulations are also known for the graphs which may be obtained as 1-sums of their members. We conclude our review on polyhedral works related to generalizations of the domination concept with a very special case: when f v = d G v , for all v ∈ V , a node subset S ⊆ V is an f -dominating set if and only if its complementary set V \S is a stable set. This implies that for this case it is equivalent to look for a maximum weight stable set and a minimum weight f -dominating set. From results on the stable set problem, it follows that [M W f ] can be solved in polynomial time in perfect graphs for the particular case when f v = d v , for all v ∈ V , and that also a complete description of D f G is known for that case (Chvátal 1975;Padberg 1974).
To the best of the authors' knowledge no polyhedral results appear in the literature on the polyhedral structure of D f G for general f ∈ F G . The only works on the problem [M W f ] for general f ∈ F G we are aware of essentially focus on the case c = 1 and establish bounds on the optimal objective value of [M W f ] for this particular case (Chen and Zhou 1998;Stracke and Volkmann 1993;Zhou 1996).

Our contribution
In this paper we introduce different results related to the polyhedral structure of the polytope D f G . They namely include different families of facet-defining inequalities and a complete description of this polytope for trees. In general, this formulation may contain an exponential number of constraints (whereas it is linear in the number of nodes for D 1 G ), but we show that the corresponding separation problem can be solved in polynomial time, thus implying the polynomial time solvability of [M W f ] for trees, by the well-known results on the polynomial equivalence between optimization and separation established by Grötschel et al. (1981). In addition, we provide a linear time combinatorial algorithm for that case, thus extending some results in the literature on the classical domination concept.

Structure of the paper
In Sect. 2, we introduce different polyhedral results on D f G among which some families of facet-defining inequalities. In Sect. 3, we provide a complete description of the polytope D f G for the case of trees and show that the problem [M W f ] can be solved in polynomial time in that case. Finally, in Sect. 4, we present a linear time algorithm for the case that G is a tree.

Additional notation
Before we close this section, let us mention some additional notation that will be used later. Given a graph G = (V , E) and a node subset S ⊆ V , let G[S] denote the subgraph of G that is induced by S, i.e., G[S] = (S, E ), where E stands for the subset of edges in E having both endpoints in S. Given a positive integer n, I n stands for the identity matrix with order n. We will also use 0 to denote a vector whose dimension will be clear from the context and having all its entries equal to zero.

Polyhedral results on the polytope D f G
In this section, we assume that G = (V , E) is a simple connected graph with n ≥ 2. After some basic polyhedral properties are introduced, we present nontrivial facetdefining inequalities of D f G .

Basic polyhedral properties of D f G
In what follows, the inequalities of the form x v ≥ 0 and x v ≤ 1 with v ∈ V , will be called trivial inequalities.
Proposition 1 Let f ∈ F G . Then the following holds.
Proof The statements (i)-(iii) and the first part of statement (iv) follow from the work by Hammer et al. (1975). So we prove the last part of (iv) (which starts from "Moreover"). Also, for property (ii), we provide a simple and short alternate proof.
Thus, in both cases, the inequality x v ≥ 0 cannot define a facet.
[⇐] The incidence vectors of the n f -dominating sets: V \{v} and V \{v, w}, for all w ∈ V \{v}, are affinely independent and they all belong to F 0 v . Proof of the last part of (iv): b > 0 and |V a | ≥ 2. The property b > 0 is implied by the fact that the inequality is not trivial (otherwise it would be dominated by the nonnegativity inequalities). If we had V a = {v}, the inequality would correspond (up to multiplication by a positive scalar) to an inequality of the form ∈ D, thus implying b = 0, and the inequality would be trivial.
Let a t x ≥ b denote a non trivial facet-defining inequality of D f G , and let G a = G[V a ] be the subgraph of G that is induced by the node subset V a . The following property was shown to hold for the dominating set polytope (Bouchakour and Mahjoub 1997), and its extension to f -dominating set polytopes is straightforward.
Proposition 2 (Bouchakour and Mahjoub 1997) The graph G a is connected.

Non trivial facet-defining inequalities
We start with a definition that will be useful for presenting different families of inequalities that are valid for D f G . Definition 1 Given an undirected simple graph G = (V , E) and f ∈ F G , an f-clique is a node subset Q ⊆ V , such that |Q| ≥ 2 and satisfying the following two conditions: The next three propositions deal with cases when some nodes of the graph satisfy then the following holds: D ⊆ V is an f -dominating set iff D is a vertex cover. So in that case, the vertex cover and f -dominating set polytopes coincide (and are both affinely equivalent to the stable set polytope). The next proposition directly follows from this correspondence, see, e.g., (Nemhauser and Trotter 1974).

Proposition 3 Let the graph C = (V (C), E(C)) be an odd cycle such that each edge {u, w} ∈ E(C) is an f -clique. Then, the inequality
(A chord is an edge joining two nonconsecutive nodes of the cycle).
The next two propositions illustrate the fact that, even though the property that " f u = d u or f v = d v for each edge {u, v} ∈ E" may hold only on some parts of the graphs, some well-known families of inequalities valid for the vertex cover polytope, may, under some conditions, define facets of f -dominating set polytopes. We now formulate a simple sufficient condition for an inequality of type (1) corresponding to a node-induced subgraph of an arbitrary graph to define a facet of D f G .
Consider now some node k ∈ V \V (C). And let S ⊂ V (C) denote a subset of exactly |V (C)| 2 nodes of the cycle C such that N (k) ∩ V (C) ⊆ S, and the node subset U = (V \V (C)) ∪ S is an f -dominating set. The existence of S follows from the assumption that |N (k) ∩ V (S)| ≤ 2. Since U and U \{k} correspond to f -dominating sets in G whose incidence vectors satisfy (1) with equality, we deduce a k = 0.
It follows that the inequality a t x ≥ b must correspond, up to multiplication by a positive scalar, to inequality (1).

Proposition 5 Let Q ⊆ V denote an f -clique. Then, the following inequality is valid for
and it is facet-defining if and only if Q is maximal (w.r.t. inclusion).
Proof The validity of (2) easily follows from the domination requirements. We now establish the necessary and sufficient conditions for it to be facet-defining.
[⇒] Assume that the inequality (2) is facet-defining. If Q is not maximal, there exists some f -clique Q such that Q ⊂ Q . But then the inequality (2) is the sum of the f -clique inequality Assume firstly that there exists some node v in Q that is not a neighbor of w. Then, since V \{v} and V \{v, w} are f -dominating sets in G whose incidence vectors belong to F, we deduce a w = 0.
Consider now the case of a node w ∈ V \Q such that Q ⊆ N (w). Since Q is a maximal f -clique, there must exist some node z ∈ Q satisfying f z < d z and we must have f w < d w . Then, since V \{z} and V \{z, w} are f -dominating sets in G whose incidence vectors belong to F, we deduce a w = 0.
For any pair (v, w) ∈ Q 2 , with v = w, since the sets V \{v} and V \{w} are f -dominating in G, we deduce a v = a w .
It follows that the inequality a t x ≥ b corresponds, up to multiplication by a positive scalar, to (2).
We now introduce another family of valid inequalities for D f G that we will call partial neighborhood inequalities, a denomination that is suggested by their support which is a subset of a closed neighborhood of a node having a positive domination requirement. These inequalities may be seen as a generalization of the classical neighborhood inequalities (obtained by setting q = f v = 1 in (3) given hereafter) used in linear formulations of the dominating set problem. Recall that these inequalities, together with the trivial inequalities, are sufficient to completely describe the dominating set polytope ( f v = 1, for all v ∈ V ) in the case of strongly chordal graphs (Farber 1984). Their relevance will be further stressed later, when considering complete formulations of the f -dominating set polytope for trees.
Proposition 6 Let G = (V , E) denote an undirected simple graph, let f ∈ F G and u ∈ V such that 1 ≤ f u < d u . Then, the following partial neighborhood inequality is valid for D f G : Proof Any f -dominating set D not containing node u must contain at least f u of its neighbors: |D ∩ N (u)| ≥ f u . Let U denote a subset of at most f u − 1 neighbors of u: U ⊂ N (u) and 0 ≤ |U | ≤ f u − 1. Then, we must have |D ∩ (N (u)\U )| ≥ f u − |U |.
With q = f u − |U |, Z q = U , we can check that the incidence vector of D satisfies (3). Also note that (3) is trivially satisfied by the incidence vector of any f -dominating set containing u.
We now formulate a simple sufficient condition for an inequality of type (3) to be facet-defining for D f G .
Proof Assume all the conditions mentioned are satisfied, fix q ∈ {1, 2, . . . , f u } and all v ∈ V \{u}, and because they both contain f u neighbors of node u), and they satisfy (3) with equality. We deduce: a w = a w .
Considering the incidence vectors of the node subsets V \N (u) and (V \N [u]) ∪ A ∪ Z q (with A as defined before), since they correspond to f -dominating sets and their incidence vectors satisfy (3) with equality, we deduce a u = qa w , with w ∈ A.
It follows that the inequality a t x ≥ b must correspond, up to multiplication by a positive scalar, to inequality (3).

Complete descriptions of D f G for trees
After we report some preliminary results in Sect. 3.1, we present a complete formulation of the f -dominating set polytope for stars in Sect. 3.2, and for trees in Sect. 3.3.

On leaves with domination requirement one
We start with an auxiliary result.

Lemma 1 Let G = (V , E) be an undirected simple graph with at least two nodes.
Let f ∈ F G and {u, w} ∈ E be such that w is a leaf node and f w = d w = 1. If the constraint a t x ≥ b is a facet-defining inequality for D f G that is not trivial and different from x u + x w ≥ 1, then, a w = 0.
Proof Let a t x ≥ b denote a facet-defining inequality satisfying the assumptions given in the statement of the lemma. Let D ⊆ V denote an f -dominating set whose incidence vector satisfies a t x = b, and such that u and w belong to D. Such an f -dominating set exists due to the fact that the inequality a t x ≥ b is assumed to be facet-defining and different from x u + x w ≥ 1. Since node w is a leaf and its only neighbor is u ∈ D, it follows that D\{w} is an f -dominating set. We deduce a w = 0. (This is due to the fact that, by Proposition 1-(iv), we have a w ≥ 0, and if we had a w > 0, then the incidence vector of D\{w} would violate a t x ≥ b, a contradiction.) Lemma 1 may be interesting when looking for a complete formulation of D f G when G contains a node w satisfying f w = d w = 1. This is illustrated by the next proposition whose proof relies on it.

Proposition 8 Let G = (V , E) be an undirected simple graph, and let f ∈ F G .
Assume that there exists a node w ∈ V such that f w = d w = 1, and let u ∈ V be the unique neighbor of w.
Then, a complete formulation of D f G is obtained by adding to that of D f G the variable x w and the inequalities: 0 ≤ x w ≤ 1 and x u + x w ≥ 1.

Proof Let a t x ≥ b be a facet-defining inequality for
Let a ∈ R n be defined as follows: a v = a v , for all v ∈ V , and a w = 0. First, notice that the inequality a t x ≥ b is valid for D Let D 1 , D 2 , . . . , D n−1 denote n − 1 f -dominating sets in G , whose incidence vectors in R V are affinely independent and satisfy the equation a t x = b. Since a t x ≥ b is assumed to be different from x u ≥ 0, there exists some index k ∈ {1, 2, . . . , n − 1} such that u ∈ D k . Then, define D j = D j ∪ {w}, for all j ∈ {1, 2, . . . , n − 1}, and D n = D k . Since the incidence vectors of the f -dominating sets (D i ) n i=1 are affinely independent and all lie in F, the inequality a t x ≥ b defines a facet of D f G . Now let g t x ≥ h be a facet-defining inequality of D f G with g w = 0, g ∈ R n \{0}. Letg denote the restriction of the vector g to its entries indexed by V . First note that the inequalityg t x ≥ h is valid for D Assume, for a contradiction, that this is not the case. Then, there exists (a, b) ∈ (R n−1 \{0} × R) such that the inequality a t x ≥ b defines a facet F of D f G satisfying F ⊂ F, and the vectors a andg are linearly independent. From the first part of the proof, the inequalityǎ t x ≥ b is facet-defining for D f G , withǎ ∈ R n \{0} defined by a v = a v , for all v ∈ V , andǎ w = 0. This impliesǎ = λg for some λ ∈ R with λ > 0, and thusg = λa, a contradiction.
Then, by Lemma 1, no other facet-defining inequality with a w = 0 exists, which concludes the proof.
Considering the last proposition, in the rest of this section we will consider graphs with all degree-one nodes having zero domination requirement, unless otherwise stated.

Complete description of D f G when G is a star
Let G = (V , E) be a star with n ≥ 3 and having for center the node 1. Let V = V \{1} and assume that 0 ≤ f 1 < n − 1, f v = 1, v ∈ V . From Proposition 8, it follows that a complete description of D f G is given by the trivial inequalities, together with the constraints x 1 + x v ≥ 1, v ∈ V . Note that this also holds for D f G with f defined as follows: For the case when f v = 0, v ∈ V , a description is given by the trivial inequalities. Together with Proposition 8, the next result leads to a complete description for the remaining cases, i.e., 0 Proof We prove that all the facet-defining inequalities of D f G correspond to trivial or type (3) inequalities.
Let a t x ≥ b denote a facet-defining inequality of D f G that is not trivial. (So, by Proposition 1-(iv), we have: |{v ∈ V : a v > 0}| ≥ 2, a ≥ 0, b > 0.) Let P a = {v : a v > 0 and v ∈ V } and Z a = v : a v = 0 and v ∈ V . Note that, since {1} is an f -dominating set, necessarily a 1 > 0, and any f -dominating set Q containing 1 and satisfying a t x = b must satisfy Q ∩ P a = ∅. Let D denote an f -dominating set such that 1 / ∈ D and whose incidence vector satisfies a t x = b. Note that, since the constraint is assumed to be valid and nontrivial, |P a | ≥ 1. Also, necessarily, Z a ⊂ D. (Otherwise, let u ∈ D such that a u > 0 and w ∈ Z a \D. The set (D\{u}) ∪ {w} would be f -dominating but its incidence vector would violate the inequality a t x ≥ b, a contradiction). This also implies |D ∩ P a | = f 1 − |Z a |. Now, let u denote a node in D that is associated with the largest coefficient value of the inequality: u ∈ argmax{a v : v ∈ D}. Since for any node w ∈ V \D, the node set (D\{u}) ∪ {w} is an f -dominating set, we deduce a w ≥ a u > 0, w ∈ V \D. Now, let u stand for a node that is different from 1, is not in D, and is associated with the largest coefficient value in the inequality: u ∈ argmax{a v : v ∈ V \D}. In particular, we have a u ≥ a u .
Let D denote another f -dominating set satisfying a t x = b, and such that u ∈ D .
(Such an f -dominating set does exist, because otherwise, the face defined by a t x ≥ b would be contained in the hyperplane x u = 0, a contradiction.) Considering the fdominating set (D \{u }) ∪ {u} for any u ∈ D\D , we deduce a u ≤ a u ≤ a u , and thus a u = a u = a w , w ∈ V \D.
Let u ∈ argmin{a v : v ∈ D\Z a }. Let D be an f -dominating set not containing u nor {1}, and such that its incidence vector satisfies a t x = b. (D exists, because otherwise all the incidence vectors of the f -dominating sets satisfying a t x = b would belong to the hyperplane with equation x 1 +x u = 1. But the latter does not correspond to a supporting hyperplane of D f G , since there exist f -dominating sets satisfying x 1 + x u = 0 and others satisfying x 1 + x u = 2). Since |D ∩ P a | = |D ∩ P a | = f 1 −|Z a | (see before), D contains some node w ∈ V \D. Then, (D \{w})∪{u } is an f -dominating set, and we deduce a u ≤ a w ≤ a u , thus implying a w = α, w ∈ V \Z a , for some constant α > 0.
Finally, considering an f -dominating set saturating the constraint and containing 1, we deduce a 1 = ( f 1 − |Z a |)α. Thus, the inequality a t x ≥ b corresponds (up to multiplication by a positive scalar), to an inequality of type (3).
The former results are summarized in the next Proposition.

Proposition 10 If the graph G is a star and f ∈ F G , then a complete description of
, by the f -clique inequalities (2) and the partial neighborhood inequalities (3).
Remark 1 By Proposition 7, for the case of a star G = (V , E) having node 1 as its center, and f ∈ F G with 1 ≤ f 1 < d 1 , the polytope D f G is full dimensional and the number of distinct facets of D f G defined by inequalities of type (3) may be exponential in n.
Next, we shall see how the results from this section may be extended to get a complete formulation for trees.

Complete description of D f G when G is a tree
Theorem 1 If the graph G = (V , E) is a tree and f ∈ F G , then a complete formulation of D f G is given by the following set of inequalities.
Proof We proceed by contradiction for the proof which relies on the results established in the former sections (Propositions 8, 10). Let P f G denote the polytope which is defined by the set of inequalities given in the statement of the theorem. Naturally, we have Let G denote a graph corresponding to a tree with a minimum number of nodes such that the result does not hold, i.e., such that the polytope P f G has a fractional extreme point x * . Note that from Proposition 10, the graph G cannot be a star. This implies n ≥ 4. Also, given the formulation of P f G , the choice of G and Proposition 8, the domination requirement of each leaf in G must be equal to zero.
Let E denote a nonsingular subsystem of n equations defining x * . Each equation of the subsystem corresponds to an inequality arising in the description of P f G . In general, this subsystem is not uniquely defined. For our purposes, we will consider such a subsystem of n equations defining x * and having a maximum number of equations corresponding to trivial inequalities. So, for each integral component x * v (if there exists one), the equation x v = 0 or x v = 1 appears in E. Note that, necessarily, at least one equation corresponding to an inequality of type (3) must be contained in E (since otherwise, the matrix whose entries correspond to the left-hand side of the subsystem E is totally unimodular, and we get a contradiction with the fact that x * is not an integer vector). And at least two variables in the support of this inequality must have positive fractional values (by the definition of the subsystem E).
We then make use of the following auxiliary claims whose proofs are postponed, for clarity. The basic idea behind those claims is to identify some properties of the fractional solution x * and of the graph G that would correspond to a minimal (w.r.t. the number of nodes) counterexample to the statement of the theorem. Their proofs heavily rely on this minimality assumption.
The first claim provides the information that the subsystem E cannot contain an equation corresponding to an f -clique inequality involving a leaf node.
Claim 1 For each leaf node w ∈ V , we have: x * u + x * w = 1, where u denotes the unique neighbor of w.
The second claim establishes that each entry of x * corresponding to a non-leaf node has value strictly less than 1.
Claim 2 For each node w ∈ V that is not a leaf, we have: x * w < 1.

The third claim states a property that is satisfied by any inequality of the type (3) which belongs to the subsystem E.
Claim 3 Assume the system E contains one equation corresponding to some inequality of type (3): The Claims 4 and 5 establish properties related to the leaves. The fourth claim states that all the entries of x * corresponding to leaves must be fractional. This is used to establish Claim 5 stating that the neighbor of any leaf node must have a domination requirement strictly less than its degree.

Claim 5 G contains no leaf that is a neighbor of a node u satisfying f u = d u .
The next two claims (whose proof relies on the former) establish the existence of a node in G that has degree two and is adjacent to exactly one leaf. This point will then be used to get a contradiction and prove the theorem.
Claim 6 Each node in G is adjacent to at most one leaf.

Claim 7
The graph G has a node with degree 2 that is adjacent to one leaf.
Let u be a node in G satisfying Claim 7 and let w denote its leaf neighbor. Then necessarily f u = 1 (due to the definition of the system P f G , the fact that d u = 2 and Claim 5). Recall that x * w is fractional (by Claim 4), and by Claim 1, the subsystem E must contain an equation which corresponds to a partial neighborhood inequality with center u, namely x u + x z + x w = 1, where z is the neighbor of u that is different from w. Also note that the variable x w only occurs in this equation of E, and at least one of the two quantities among x * u and x * z must be fractional. Let E denote the subsystem obtained from E by removing this equation and the column corresponding to x w . Then E is nonsingular and all the equations that are present in E correspond to inequalities that are present in the description of P The restriction x * of x * to its components corresponding to nodes in V \{w} belongs to P f G and also satisfies E . It follows that x * corresponds to a fractional extreme point of P f G , contradicting the hypothesis that G has the minimum number of nodes. We now give the details of the proofs of the seven claims used in the proof of Theorem 1.
Proof of Claim 1 Assume, for a contradiction that there exists an edge {u, w} ∈ E such that w is a leaf and x * u + x * w = 1. Let x * denote the restriction of x * to its entries indexed on V \{w}. Let G = G [V \{w}], and define f ∈ F G as follows: f v = f v , for all v ∈ V \{u, w} and f u = f u −1. Then, one can easily check that x * ∈ P f G . And from our minimality assumption on G, x * can be expressed as a convex combination of incidence vectors of f -dominating sets in G : where q denotes a positive integer, the sets (S i ) q i=1 are f -dominating sets in G and λ ∈ R q + satisfies q i=1 λ i = 1. Let us now consider the sets (Ŝ i ) q i=1 defined as follows: It can be checked that the sets (Ŝ i ) q i=1 correspond to f -dominating sets in G. But this also implies that x * = q i=1 λ i χŜ i , contradicting the fact that x * is a fractional extreme point of P f G .
to v. For a given v ∈ V , let T v = (V v , E v ) denote the subtree of G with root v, i.e., the collection of all the paths from v to the leaves, that do not contain the root node r . Finally, let H v = {u ∈ V : σ (u) = v} be the set of immediate successors (or children) of node v.
To compute an optimum f -dominating set we use a recursive dynamic programming method which associates three labels to each node v ∈ V : 0}. The next lemma shows that we can compute these labels using a bottom-up recursion, from the leaves to the root r . Label C 1 (r ) gives the optimal solution value for the entire graph G.
Lemma 2 Given a tree G = (V , E), rooted at node r , c ∈ R n + , and a subtree T v rooted at node v, one can compute the labels C 1 (v), C 2 (v) and C 3 (v) as follows. If node v is a leaf, we have where the minimization subproblem over I in the expressions of C 1 (v) and respectively. Also the sum over an empty set of indices is assumed to take value 0.
Proof Consider a node v ∈ V . If v is a leaf, the correctness of the labels is straightforward, so let us suppose d v > 1. Let D denote a minimum cost f -dominating set in T v . Label C 1 (v) gives the value of an optimal f -dominating set for The inner minimum in (4), for each possible choice of f v successors of v, considers the optimal costs of the corresponding subtrees, plus the optimal cost for the subtrees rooted at the remaining successors. The outer minimum chooses the best option between the two above cases: v ∈ D and v / ∈ D. C 2 (v) gives the optimal solution value for T v if v ∈ D. Therefore, it accounts for c v , plus the sum of the optimal cost of each subtree rooted at a node u ∈ H v , provided that u is already dominated by v ∈ D. Therefore C 3 (u) is the required value for subtree T u . C 3 (v) is the optimal solution value for T v if the domination requirement of v is set to max{ f v − 1, 0}. The formula is similar to the one used to compute the label C 1 (v). The outer minimum chooses between the case v ∈ D (where C 3 (v) = C 2 (v) gives the optimal value), and the case v / ∈ D, where max{ f v − 1, 0} successors of v must be inserted in D. The inner minimum in (6) operates as the equivalent minimum in (4), but considering subsets of H v with cardinality max{ f v − 1, 0}.
From Lemma 2, we deduce a dynamic programming algorithm to compute both the optimal objective value of [M W f ] and an optimal solution (by storing the sets generating the minima in the definition of the labels). This is illustrated by the next example. Figure 1 shows a small example with a tree of nine nodes and root 1. Near each node v we report: (a) in square brackets, the weight c v and the domination requirement f v , respectively; (b) in parenthesis, the three labels C 1 (v), C 2 (v) and C 3 (v). When the value of C 1 or C 3 is determined by the inner minimum and is used in the second phase (described later) to determine an optimal solution, we report also the set which produces the minimum. In a first phase, the algorithm computes the labels recursively, starting from the leaves and moving toward the root, using formulas (4)-(6). The value of the minimum f -dominating set is given by C 1 (1) = 6. Then, in a second phase, we can identify a set D of nodes corresponding to an optimal solution, by starting from root 1, with D = ∅. The fact that there is no ancestor for the root, and that C 1 (1) < C 2 (1) indicate that node 1 is not in the solution. Since f 1 = 2 we know that two of its successors have been used when computing the inner minimum in (4). Set I 1 (1), stored in the first phase, contains the two nodes giving the minimum, namely nodes 2 and 3. We set D = {2, 3} and we consider each of these nodes in turn. The fact that they have been fixed in the solution implies that labels C 2 (2) and C 2 (3) have been chosen by the recursion for these nodes, and using (5) we know that labels C 3 ( j) have been chosen for j = 5, . . . , 9. All these labels have value zero, so none of these nodes is in the solution. It remains to consider node 4. Its ancestor, the root, is not in D and C 1 (4) < C 2 (4), so the node is not in solution.

Example 1
The next result directly follows and extends the one presented by Farber 1984 showing [M W 1 ] can be solved in linear time for trees.

Theorem 2 If G = (V , E) is a tree rooted at r , then problem [M W f ] can be solved in linear time.
Proof Lemma 2 shows that label C 1 (r ), computed using (4)-(6), gives the optimal solution value. The second phase of the dynamic programming algorithm retrieves the nodes of the set D which produce the value C 1 (r ), i.e., an optimal set. In order to complete the proof, it is sufficient to show that the labels can be computed in time O(d v ), for each node v ∈ V . This is trivial for the case when v is a leaf or f v = 0. So, let v denote a node that is not a leaf and such that f v > 0. Observe that the argument of the inner minimization in (4) and (6) can be rewritten as Let S 1 (resp. S 2 ) denote the node subset of H v corresponding to the f v (resp. f v − 1) smallest values in the set C = {C 2 (u) − C 1 (u) : u ∈ H v }. It can be easily checked that S 1 (resp. S 2 ) corresponds to an optimal solution for the minimization problem over I in the expression of C 1 (v) (resp. C 3 (v)).
Using the results by Blum et al. 1973, the f v -th smallest number w in C can be found in time O(|H v |). Then, iterating (in an arbitrary order) over H v , putting into S 1 all the nodes corresponding to a value smaller than w, and then filling up S 1 with nodes associated with the quantity w to have |S 1 | = f v , we deduce that the inner minimization problem in (4) can be found in time O(d v ). The argument to prove that C 3 (v) can be determined within the same time complexity is analogous.

Conclusions
In this paper, we presented descriptions of the f -dominating set polytope when the graph is a star or a tree. They namely lead to the polynomial time solvability of the problem [M W f ] for these graphs. In addition, we presented a linear time algorithm for this problem on trees. Further research work will be directed towards extensions of these results for other graph families. The development of efficient methods and further investigations on the polyhedral structures of polytopes related to generalizations of other variants of domination are under work.