Equistarable bipartite graphs This research is supported in part by Agencija za raziskovalno dejavnost Republike Slovenije, research program P1–0285 and research projects J1-5433, J1-6720, J1-6743, and BI-US/14–15–050. The first author also thanks for partial support the National Science Foundation (Grant IIS-1161476).

Equistarable bipartite graphs This research is supported in part by “Agencija za raziskovalno dejavnost Republike Slovenije”, research program P and research projects J-, J-, J-, and BI-US/. The first author also thanks for partial support the National Science Foundation (Grant IIS-1161476).

Abstract

In this paper we characterize equistarable bipartite graphs. We show that a bipartite graph is equistarable if and only if every -matching of the graph extends to a matching covering all vertices of degree at least . As a consequence of this result, we obtain that Orlin’s conjecture holds within the class of complements of line graphs of bipartite graphs.

We also connect equistarable graphs to the triangle condition, a combinatorial condition known to be necessary (but in general not sufficient) for equistability. We show that the triangle condition implies general partitionability for complements of line graphs of forests, and construct an infinite family of triangle non-equistable graphs within the class of complements of line graphs of bipartite graphs.

Keywords: equistable graph, general partition graph, bipartite graph, equistarable graph, graph
MSC (2010): 05C69, 05C50, 05C22, 05C76

1Introduction

In [13] Milanič and Trotignon established a connection between equistarability and equistability and between -extendable graphs and general partition graphs. (See Section 2 for definitions.) In particular, they proved that: (1) if a graph is triangle-free, then is equistarable if and only if , the complement of its line graph, is equistable and (2) a connected triangle-free graph of minimum degree at least is -extendable if and only if is a general partition graph. Based on this approach, they disproved Orlin’s conjecture [12], which stated that every equistable graph is a general partition graph. The counterexamples in [13] are based on complements of line graphs of triangle-free graphs, and that work left open the validity of Orlin’s conjecture for the class of complements of line graphs of bipartite graphs, and more generally for the class of perfect graphs. Regarding other subclasses of perfect graphs, Orlin’s conjecture can be easily verified to hold for bipartite graphs and their complements, and was shown to hold for chordal graphs [15] and for line graphs of bipartite graphs [8].

We show in this paper that Orlin’s conjecture holds for the complements of line graphs of bipartite graphs. We achieve this by further extending the connections between a triangle-free graph and the complement of its line graph , translating the properties that is a general partition graph, resp. a triangle graph, to . We recall that the general partition property and the triangle property are a sufficient and a necessary condition, respectively, for equistability. We summarize the connections in Table ?.

We show that, when restricted to the class of bipartite graphs, the upper three classes on the left in Table ? coincide. Moreover, when restricted to the class of forests, all four classes on the left in Table ? coincide.

The paper is structured as follows. Section 2 contains the basic definitions and known lemmas that establish the known equivalences and implications in Table ?. In Section 3 we establish the first and the last equivalence from Table ? and observe that the implications in the left side of the table cannot be reversed. Sections Section 4 and Section 5 contain our main results, that is, a complete characterization of equistarable bipartite graphs and of equistarable forests, along with some algorithmic aspects concerning the recognition of these newly characterized families.

2Preliminaries

All graphs in the paper will be finite, simple and undirected. For undefined graph theoretic notions, we refer to [19]. A stable or (independent) set in a graph is a set of pairwise non-adjacent vertices; a stable set is said to be maximal if it is not contained in any other stable set. A clique in a graph is a set of pairwise adjacent vertices. We denote by the set of all neighbors of and . The degree of a vertex in a graph , denoted by , is equal to . The minimum degree of a graph is the minimum degree of its vertices and is denoted by .

The complement of a graph is the graph with the same vertex set as in which two distinct vertices are adjacent if and only if they are not adjacent in . The line graph of is a graph such that: the vertex set of is the edge set of and two distinct vertices of are adjacent if and only if they share a common endpoint as edges in . A graph is bipartite if its vertex set can be partitioned into two independent sets, and triangle-free if it does not have a triangle () as induced subgraph. A graph is said to be a star if it is isomorphic to the complete bipartite graph for some .

A graph is a general partition graph if there exists a set and an assignment of non-empty subsets to the vertices of such that two vertices and are adjacent if and only if and for every maximal stable set of , the set is a partition of [11]. A graph is said to be equistable if and only if there exists a mapping such that for all , is a maximal stable set in if and only if [14]. In 1994 Mahadev et al. introduced in [10] a subclass of equistable graphs, the so-called strongly equistable graphs. For a graph , we denote by the set of all maximal stable sets of , and by the set of all other nonempty subsets of . A graph is said to be strongly equistable if for each and for each there exists a function such that for all , and . The triangle condition was introduced by McAvaney et al. in [11] and states that for every maximal stable set in and every edge in there is a vertex such that induces a triangle in . Graphs satisfying this condition are called triangle graphs.

A matching in a graph is a set of pairwise disjoint edges. Given a matching in a graph , we say that a vertex is covered (or saturated) by if is an endpoint of an edge in . We will denote by the set of all endpoints of edges in . Given two matchings and in a graph , we say that extends to if . A matching consisting of exactly edges will be referred to as a -matching (where is the size of the matching). A matching is said to be a perfect matching of if it covers all vertices of . We say that a matching in a graph is a perfect internal matching if every vertex not covered by is a leaf, that is, a vertex of degree 1. Perfect internal matchings were studied in a series of papers, see for example [1] and references cited therein. For a general reference on matching theory, see [9].

A connected graph is said to be -extendable if contains a -matching and every -matching can be extended to a perfect matching [16]. We generalize this notion as follows.

Even though for the purposes of stating the main results in this paper (Theorems ? and ?), it would be more convenient to define a graph to be -internally extendable simply as a graph in which every -matching extends to a perfect internal matching, we decided to keep the definition more restrictive, requiring also connectedness and the existence of a -matching. This is because this way, the definition is similar to the definition of -extendable graphs; moreover, the two notions coincide for graphs of minimum degree at least . In this paper, we consider -internally extendable graphs only for .

Given a graph and a vertex , the star rooted at is the set of all edges incident with . A star of is a star rooted at some vertex and a star is said to be maximal if it is not properly contained in any other star. We denote the union of stars from a set of vertices as , where . In [13] equistarable graphs were introduced as graphs without isolated vertices for which there exist a mapping on the edges of such that a subset is a maximal star in if and only if . Such a mapping is called an equistarable weight function of . Note that for every equistarable weight function , we have for all , since otherwise that would directly imply that we would have a total weight of on some subset of the edges that does not induce a maximal star in . For a graph we denote by the set of all maximal stars of , and by the set of all other nonempty subsets of . A graph without isolated vertices is said to be strongly equistarable if for each and each there exists a mapping such that for all , and [13]. Given a graph and a subset of edges , the characteristic vector of is the vector defined as if , and , otherwise.

We also introduce the following property imposing a constraint on -vertex paths in the graph. A graph is said to be -constrained if the middle vertex of every (not necessary induced) -vertex path in is of degree at least , that is, it is incident with at least one edge not in the path.

We conclude this section by discussing the validity of Table ?. The following lemmas from [13] establish the second and third equivalences in Table ?.

The first and the third implication in the right side of Table ? were proved by Miklavič and Milanič in [12] (the third one was essentially observed already in [10]), and, as already mentioned, the second implication in the right side of Table ? was proved by Mahadev et al. in [10]. Lemmas ? and ? along with the second implication in the right side of Table ? directly imply the second implication in the left side of Table ?.

In fact, a straightforward adaptation of either the geometrical proof of the result of Mahadev-Peled-Sun from [10] or the alternative proof given in [13] to the setting of strongly equistarable graphs shows that the same statement holds for general graphs (not necessary triangle-free). That is, every strongly equistarable graph is equistarable.

The first and the fourth equivalence in Table ? will be proved in Lemmas ? and ?, respectively, in Section 3. In turn, this will imply the validity of the first and the third implication in the left side of Table ?.

3Basic results and examples

As proved by Korach et al. in [7], a graph is equistable if and only if each connected component of is equistable. As we show in Lemma ? below, the analogous property for equistarable graphs holds only in one direction. To this end, recall that in [13] the following property of the graph depicted in Figure 1 was observed.

Figure 1: A graph H that is equistarable but not strongly equistarable.
Figure 1: A graph that is equistarable but not strongly equistarable.

The statement of Lemma ? appeared within the proof of Proposition 2 in [13] (where graph was named ) and was established using the structure of a basis of the kernel of the incidence matrix of the graph. We offer an alternative, shorter proof, using a method that we will apply again later (in the proof of Proposition ?).

Let be the mapping as depicted in Figure 2.

Figure 2: The mapping \lambda:V(H)\to\{-1/2,1/2\}.
Figure 2: The mapping .

That is, vertices in the bigger part of the bipartite graph get assigned weight , and all the other vertices get weight . Using the coefficients given by the mapping , the characteristic vector of the -matching can be expressed as a linear combination of the characteristic vectors of maximal stars, that is, Since and for all , this implies that .

If is an equistarable weight function of a graph and is a connected component of , then , the restriction of to is an equistarable weight function of . This follows immediately from the definitions, using the fact that for every subset of , is a maximal star of if and only if it is a maximal star of .

The fact that the class of equistarable graphs is not closed under taking disjoint union can be now justified using Lemma ?. Indeed, the lemma implies that the disjoint union of two copies of contains a -matching of total weight in each equistarable weight function, and is hence not equistarable.

The following consequence of Lemma ? is in contrast with the fact that the class of strongly equistable graphs is closed under join, as proved by Mahadev et al. in [10].

Since the graph depicted in Figure 1 is an equistarable triangle-free graph, the complement of its line graph is equistable (see Table ?). Let denote the disjoint union of two copies of . It can be seen that the graph is isomorphic to the join of two copies of . However, since the graph is not equistarable, the graph is not equistable.

The following lemma is from [13].

We now generalize Lemma ?, thus establishing the first equivalence in Table ?. In the proof we will make use of a result by McAvaney et al., which we now state. A strong clique in a graph is a clique containing at least one vertex (equivalently: exactly one vertex) from each maximal stable set.

We first show the equivalence of conditions 1 and 2, and then the equivalence of conditions 2 and 3.

For , we first use Theorem ? to infer that it suffices to show that every edge of belongs to a strong clique if and only if every -matching of extends to a perfect internal matching. It follows from the definitions of the complement and the line graph operators that a subset is a -matching of if and only if it forms an edge of . Moreover, a set forms a strong clique in if and only if forms a stable set in intersecting all maximal cliques of . Since is triangle-free, maximal cliques of correspond bijectively to the maximal stars of . Therefore, a stable set in intersecting all maximal cliques of is a matching of intersecting all maximal stars of . Equivalently, is a perfect internal matching of that in addition contains all unique edges of components of isomorphic to (since the stars rooted at any vertex of the are maximal stars).

The above implies that every edge of belongs to a strong clique if and only if every -matching of extends to a perfect internal matching that contains all unique edges of components of isomorphic to . This last condition is easily seen to be equivalent to the condition that every -matching of extends to a perfect internal matching, and establishes the equivalence .

To prove , suppose that every -matching of extends to a perfect internal matching, let be a component of that is not a star, and let be a -matching in . By assumption, extends to a perfect internal matching . Then, matching is a perfect internal matching of extending .

It remains to show . Suppose that every connected component of is either a star or -internally extendable. Then, each component of contains a perfect internal matching, say . Let be a -matching in . If is contained in a single connected component of , say , then is contained in a perfect internal matching of , say , and a perfect internal matching of extending is given by , where denotes the set of all connected components of .

Suppose now that the two edges of belong to different connected components of , say and . We claim that is contained in an internal perfect matching of (and then by symmetry, is contained in an internal perfect matching of ). If extends to a -matching of , then we can apply the assumption that is -internally extendable, and the claim follows. So suppose that is a maximal matching in . Then, the set , where , is an independent set in . Since is triangle-free, every vertex of is of degree in . But this implies that itself is a perfect internal matching of .

Denoting by and internal perfect matchings of and containing and , respectively, a perfect internal matching of extending is given by . This establishes the implication and completes the proof.

The last equivalence from Table ? is proved in the next lemma. Recall that a graph is said to be -constrained if the middle vertex of every (not necessary induced) -vertex path in is incident with at least one edge not in the path.

Suppose first that satisfies the triangle condition but is not -constrained. Therefore there exists a in where . The set is a maximal star in , therefore in it is a maximal stable set, because is triangle-free. Let us introduce and . Since in the graph the vertex is adjacent to but not to and is adjacent to but not to , we conclude that the edge in can not be extended to a triangle with a vertex from the maximal stable set , a contradiction with the triangle condition.

Suppose now that is -constrained. We will verify that the triangle condition holds for . Take a maximal stable set in and a pair of adjacent vertices . Note that is a maximal star in , centered at some vertex and is a -matching in . Note also that since , these two edges are not incident with in . It is enough to show that there exists an edge such that form a -matching in . Equivalently, we need to show that has a neighbor not covered by the matching . If , that is, , then, since is a maximal star, the connected component of containing vertex is a single edge, . Thus, no edge of is incident with , and hence is a -matching. If , then, since is triangle-free and -constrained, at most one neighbor of is covered by , hence in this case there exists an uncovered neighbor of . Similarly, if , then the triangle-freeness implies that and cover at most two neighbors of , and again there exists an uncovered one. So in all cases we can extend to a -matching using an edge incident with . This shows that satisfies the triangle condition.

3.1Counterexamples to converses of the implications in Table

As explained at the end of Section 2, Lemmas ? and ? together with previously known results suffice to justify all implications and equivalencies in Table ?. In [12], circulant graphs of the form for odd were given as examples of a connected triangle-free strongly equistarable graphs that are not -extendable; thus, since -regular, these graphs are also not -internally extendable. This shows that the first implication on either side of Table ? cannot be reversed. The graph depicted in Figure 1 was given in [12] as an example of an equistarable but not strongly equistarable graph. Note that the graph is triangle-free, and hence shows that the second implication on either side of Table ? cannot be reversed.

What about the third implication on either side of Table ?? Regarding examples of non-equistable graphs satisfying the triangle condition, all the examples known so far (to us) can be found in [12]. These are:

  • A specific example discovered already by DeTemple et al. [5]: a -vertex graph given by and by the family of its maximal stable sets , .

  • An infinite family consisting of tensor product graphs of the form with .

(Recall that the tensor product of two graphs and is the graph with vertex set in which two vertices and are adjacent if and only if and .) It is an easy exercise to verify that for each , we have where denotes the graph isomorphism relation. Therefore, since the graphs are triangle-free, Lemma ? implies that the graphs for are -constrained and non-equistarable. In the next proposition, we offer a short direct proof of this fact.

Let with . Since the graph is -constrained. Suppose is equistarable and fix an equistarable weight function . Note that every vertex of is a center of a maximal star. Hence, , a contradiction.

The specific example mentioned above was introduced by DeTemple et al. in [5] as the intersection graph of a system of chords on a circle (represented on the left in Fig. ?).

Figure 3: Left: a system of chords in a circle, the intersection graph of which is G^* = \overline{L(K_{2,3}^+)}; right: K_{2,3}^+.
Figure 3: Left: a system of chords in a circle, the intersection graph of which is ; right: .

It turns out that is also of the form where is a bipartite -constrained non-equistarable graph. In fact, such is the smallest member of an infinite family of bipartite -constrained non-equistarable graphs, which we define now. Let be the complete bipartite graph with a fixed bipartition of its vertex set with and . Let be a graph obtained from by adding to each vertex from set a private neighbor (a leaf). Then . The graph is shown in Fig. ?.

Let , with bipartition such that , where is the set of leaves and , . Since , we do not have any vertices of degree , so is -constrained. Suppose that is equistarable and fix an equistarable weight function . Note that every vertex of is the center of a maximal star. Summing up the maximal stars from each partition we get implying . Since for all , we have and therefore , which together with implies and hence . This is a contradiction since the set of edges in does not induce a star in .

All the above examples of -constrained non-equistarable triangle-free graphs are bipartite. In the next proposition we exhibit a graph that is not of this form.

Let be the Petersen graph. Clearly, is triangle-free as well as -constrained (since it is -regular). Suppose for a contradiction that is equistarable and let be an equistarable weight function of . Fix a -matching of such that the subgraph of induced by is -regular (for example, let consist of the three thick edges as in Figure 4), and let be the mapping given by vV(M)

Figure 4: The Petersen graph together with a 3-matching and a \{1/2, -1/2\}-weighting of its vertices
Figure 4: The Petersen graph together with a -matching and a -weighting of its vertices

To complete the proof, observe that the characteristic vector of can be expressed in the form

However, since , this means that we have expressed the characteristic vector of as an affine combination of the characteristic vectors of the (maximal) stars of . Since for all , this implies that , contrary to the fact that is an equistarable weight function of and is not a star. This shows that the Petersen graph is not equistarable.

4Equistarable bipartite graphs

When restricted to complements of line graphs of triangle-free graphs of minimum degree at least , Orlin’s conjecture can be rephrased in terms of equistarable graphs as follows: every connected component of an equistarable triangle-free graph of minimum degree at least is -extendable. As the graph in Figure 1 shows, this is not the case. The work [13] left open the validity of Orlin’s conjecture for the class of complements of line graphs of bipartite graphs, and more generally for the class of perfect graphs. In this section, we prove that Orlin’s conjecture holds for complements of line graphs of bipartite graphs (note that these graphs are perfect), using the notions of - and -internal extendability (see Definition ?). In particular, we show that in the case of bipartite graphs, the classes of graphs in which each connected component is either a star or -internally extendable, strongly equistarable graphs, and equistarable graphs, all coincide (cf. Table ?).

We offer two proofs of this lemma. Our first proof is based on the classical Birkhoff-von Neumann theorem on doubly stochastic matrices.

Fix a bipartition of , and an equistarable weight function . Since , every star is maximal. It follows that . Let , and let be the matrix with rows are indexed over , and columns indexed over defined by

For every , we have , and similarly for all . Since only has non-negative entries, it is doubly stochastic. By the Birkhoff-von Neumann theorem, can be written as a convex combination of permutation matrices, say where for all and .

To show that is -extendable, we need to argue that every edge is contained in a perfect matching. Note that (since otherwise would be a set of edges of unit -weight not equal to a star). Consequently, , so there exists some such that and . We claim that is a perfect matching in . To see this, it suffices to show that . But this follows from the fact that implies . We conclude that is -extendable.

Our second proof of Lemma ? is derived using the following characterization of -extendable bipartite graphs.

In the proof of Lemma ? below, we will make use of the following result on matchings.

Let be an equistarable bipartite graph and let be a component of . Then is equistarable by Lemma ?. Fix a bipartition of , and let be an equistarable weight function of .

Suppose that is not a star. Let be the set of leaves of and let us say that a vertex is internal if it is neither a leaf nor adjacent to a leaf.

We split the proof into two cases.

Case 1:

.

Since the graph does not have any leaves, Lemma ? implies that is -extendable. We claim that is also -extendable. Suppose that is not the case, and let with and be a -matching not contained in any perfect matching. By Hall’s Theorem, there exists a subset such that . Since the graph is -extendable, must have edges to both and , and we must have .

It follows that the sum of the characteristic vectors of the stars corresponding to minus the sum of the characteristic vectors of the stars of vertices in defines a subset of the edges of such that ; in particular, is not a star. Since , we have . This contradicts the fact that is an equistarable weight function of . Thus in this case is -extendable, which is equivalent to the condition that is -internally extendable.

Case 2:

.

We start by proving that for every subset of internal vertices in (or in ) we have . Since each vertex in is the center of a maximal star, we have and therefore we just have to rule out the cases when .

If , then since is connected, and (since in this case), there must exist an edge incident with but not with . Therefore, since is strictly positive on all edges, we get , a contradiction. Suppose now that for some subset of internal vertices, and let . Clearly, . Since is an equistarable weight function of , the set is a maximal star. Since every vertex in incident with an edge in is also incident with an edge not in , the star cannot be rooted at a vertex in . Therefore, it is rooted at some vertex with . Since the star is maximal, we have . By connectedness, this implies that and . In particular, all vertices of are of degree at least . Therefore, has at least one leaf in the set , which is in contradiction with the assumption that consists of internal vertices only. Therefore we have shown that .

Now let be a -matching of . The inequality we have proven implies that for every subset of internal vertices in (or in ), we have . Therefore, by Hall’s theorem there exists a matching in covering all internal vertices in and a matching in covering all internal vertices in . By Theorem ?, graph contains a matching covering all internal vertices in . This matching together with covers all internal vertices in . Some vertices incident with a leaf might still be uncovered, but they can be covered one by one with new edges to form a perfect internal matching containing . This proves that is -internally extendable and hence the lemma is proved.

Now we have everything ready to prove the characterization of equistarable bipartite graphs,

Since bipartite graphs are triangle-free, we already know (a) (b) (by Lemma ?) and (a) (c) (d) (by Table ?). Lemma ? establishes the implication (d) (a) completing the proof of the theorem.

Table ? and Theorem ? imply the following.

Lemma ? and its proof show that triangle-free equistarable graphs are not closed under taking disjoint union. On the other hand, Theorem ?, in particular the equivalence of items (a) and (d), shows that for the case of bipartite graphs, we have the following.

We conclude this section with an algorithmic remark. To the best of our knowledge, the computation complexity status of recognizing graphs in each of the following classes is open: (strongly) equistable graphs, general partition graphs, triangle graphs, (strongly) equistarable graphs. The characterization of equistarable bipartite graphs given by Theorem ? implies an efficient recognition algorithm for this special case.

Given a graph , testing for bipartiteness can be done in linear time, and we may also assume that has no isolated vertices. By Theorem ?, in order to determine if is equistarable, it is enough to enumerate all of the -matchings of , and check, for each of them, if it can be extended to a perfect internal matching. Given a -matching , this can be done as follows. Defining , the problem becomes equivalent to the problem of determining if the graph contains a matching covering all vertices in . This problem can be solved in polynomial time either via matching matroids (see, e.g., [18]), or by reducing the problem to an instance of the maximum weight matching problem. This can be done by assigning a weight to each edge of the graph . In this case, the graph contains a matching that covers if and only if the graph contains a matching of total weight . Since the maximum weight matching problem is solvable in polynomial time [6], the proposition is proved.

5Equistarable forests

Recall that a forest is an acyclic graph. In this section we show that the classes of forests in which each connected component is -internally extendable or a star, strongly equistarable forests, equistarable forests, and -constrained forests, all coincide (cf. Table ?).

The proof is by induction on the number of vertices. For the statement is true, since the tree with two vertices has only one edge. Let , let be a tree on vertices and let be the edge of that we want to extend into a perfect internal matching. Suppose the statement of the lemma is true for every tree on at most vertices. We now show that it holds also for . The removal of the endpoints of (along with all the edges incident to them) from results in a forest with connected components , . Let . For each , a tree has no more than vertices (and at least one edge) and is therefore -internally extendable by the inductive hypothesis. For each such , let be the unique vertex in adjacent to an endpoint of , and let be an edge in incident with . Let be a perfect internal matching of that extends . By our inductive hypothesis, exists. Using the fact that every leaf of is also a leaf of one of the subtrees, we can construct a perfect internal matching of containing as: . Therefore we conclude that is -internally extendable.

Using the above lemma, we now prove the stated characterization.

The fact that (a), (b), (c), (d) are equivalent follows from Theorem ?. Since forests are triangle-free, implication (d) (e) is given by Table ?.

It remains to show (e) (a). Let a forest be -constrained. Let be a connected component of . Then, is also -constrained. Suppose that is not a star. Fix a -matching of and consider the (unique) shortest path in between and . We construct another matching by putting in it for every vertex of not covered by , an arbitrary edge incident with it and not in . (Since is -constrained, all the vertices of have degree at least .) By deleting all edges in , we are left with a forest consisting of trees (note that every leaf of is also a leaf in ), each of which contains exactly one edge of . By Lemma ?, this edge can be extended to a perfect internal matching in the corresponding tree. This means that the matching can be extended to a perfect internal matching of , thus is -internally extendable.

Since forests are bipartite graphs, testing whether a given forest is equistarable can be done in polynomial time by Proposition ?. Theorem ? implies that this can be done even in linear time.

It is well know that acyclicity can be tested in linear time using depth-first search. Given a forest , in order to determine if is equistarable, it is enough to check whether each connected component of is -constrained (by Theorem ?). This means that it is enough to identify all vertices with and check that each of them has at least one leaf in their neighborhood.

Acknowledgement

The third author is grateful to Nicolas Trotignon and Denis Cornaz for stimulating discussions on the topic.

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