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We investigate the terminal-pairability problem in the case when the base graph is a complete bipartite graph, and the demand graph is a (not necessarily bipartite) multigraph on the same vertex set. In computer science, this problem is known as the edge-disjoint paths problem. We improve the lower bound on the maximum value of $\Delta(D)$ which still guarantees that the demand graph $D$ has a realization in $K_{n,n}$. We also solve the extremal problem on the number of edges, i.e., we determine the maximum number of edges which guarantees that a demand graph is realizable in $K_{n,n}$., Comment: 15 pages, draws from arXiv:1702.04313

In this paper, we introduce the notions of m-shadow graphs and n-splitting graphs,m ≥ 2, n ≥ 1. We prove that, the m-shadow graphs for paths, complete bipartite graphs and symmetric product between paths and null graphs are odd graceful. In addition, we show that, the m-splitting graphs for paths, stars and symmetric product between paths and null graphs are odd graceful. Finally, we present some examples to illustrate the proposed theories.

The diamond is the graph obtained by removing an edge from the complete graph on 4 vertices. A graph is (\(P_6\), diamond)-free if it contains no induced subgraph isomorphic to a six-vertex path or a diamond. In this paper we show that the chromatic number of a (\(P_6\), diamond)-free graph G is no larger than the maximum of 6 and the clique number of G. We do this by reducing the problem to imperfect (\(P_6\), diamond)-free graphs via the Strong Perfect Graph Theorem, dividing the imperfect graphs into several cases, and giving a proper colouring for each case. We also show that there is exactly one 6-vertex-critical (\(P_6\), diamond, \(K_6\))-free graph. Together with the Lovasz theta function, this gives a polynomial time algorithm to compute the chromatic number of (\(P_6\), diamond)-free graphs.

The strong perfect graph theorem is the proof of the famous Berge’s conjecture that the graph is perfect if and only if it is free of odd holes and odd anti-holes. The conjecture was settled after 40 years in 2002 by Maria Chudnovsky et. al. and the proof was published in 2006. However, that proof is lengthy and intricate and using a combinatorial approach. We provide simple short proofs of the strong perfect graph theorem using polyhedral methods. We first prove the weak perfect graph theorem by polyhedral methods and use that to prove the strong perfect graph theorem. Our proofs emerge naturally from our work to calculate the capacity of multihop wireless networks. As a corollary of our proofs techniques we prove P=NP by proving there is an algorithm to find the maximum independent set or the independence number of any graph in polynomial time as a function of the number of the graph vertices.

The vertices of a graph are classified into seven types by J.T. Hedetniemi, S.M. Hedetniemi, S.T. Hedetniemi and T.M. Lewis and they ask the following questions: (1) What is the smallest order n of a graph having $$n-2$$ very typical vertices or $$n-2$$ typical vertices? (2) What is the smallest order of a pantypical graph? We answer these two questions and determine all the possible orders of the graphs in these three classes in this paper.

In this article, a technique to construct cyclic orthogonal double covers (CODCs) of regular circulant graphs by certain infinite graph classes such as complete bipartite and tripartite graphs and disjoint union of butterfly and K 1 , 2 n − 10 is introduced.

In this paper, we derive bounds on the optimal average information ratio of the access structures based on general graphs and investigate the value of the ratio for unicycle graphs and some bipartite graphs. We determine the exact values of this ratio for some infinite classes of bipartite graphs and unicycle graphs. This extends previous results. We also provide good bounds on the optimal average information ratio for all unicycle graphs.

A C k -free 2 -matching in an undirected graph is a simple 2 -matching which does not contain cycles of length k or less. The complexity of finding the maximum C k -free 2 -matching in a given graph varies depending on k and the type of input graph. The case where k = 4 and the graph is bipartite, which is called the maximum square-free 2 -matching problem in bipartite graphs, is well-solved. Previous results for this special case include min-max theorems, polynomial combinatorial algorithms, a linear programming formulation with dual integrality for a weighted version, and discrete convex structures. In this paper, we further investigate the structure of square-free 2 -matchings in bipartite graphs and present new decomposition theorems. These theorems serve as analogues of the Dulmage-Mendelsohn decomposition for matchings in bipartite graphs and the Edmonds-Gallai decomposition for matchings in nonbipartite graphs. We exhibit two canonical minimizers for the set function in the min-max formula, and a characterization of the maximum square-free 2 -matchings with the aid of these canonical minimizers.

Let G be a balanced bipartite graph of order 2 n ≥ 4 , and let σ 1 , 1 ( G ) be the minimum degree sum of two non-adjacent vertices in different partite sets of G . In 1963, Moon and Moser proved that if σ 1 , 1 ( G ) ≥ n + 1 , then G is hamiltonian. In this note, we show that if k is a positive integer, then the Moon–Moser condition also implies the existence of a 2-factor with exactly k cycles for sufficiently large graphs. In order to prove this, we also give a σ 1 , 1 condition for the existence of k vertex-disjoint alternating cycles with respect to a chosen perfect matching in G .

In this paper we characterize distance-regular graphs with diameter three having exactly three distinct distance eigenvalues, and also bipartite distance-regular graphs with diameter four having three distinct distance eigenvalues. We derive some properties and give particular examples of such graphs. We also present an infinite family of bipartite semiregular graphs with diameter four having exactly four distinct distance eigenvalues. With these results, we address some problems posed in Atik and Panigrahi (2015) [3] .

Even and odd pairs are important tools in the study of perfect graphs and were instrumental in the proof of the Strong Perfect Graph Theorem. We suggest that such pairs impose a lot of structure also in arbitrary, not just perfect graphs. To this end, we show that the presence of even or odd pairs in graphs imply a special structure of the stable set polytope. In fact, we give a polyhedral characterization of even and odd pairs.

A strong edge-coloring of a graph G=(V,E) is a partition of its edge set E into induced matchings. We study bipartite graphs with one part having maximum degree at most 3 and the other part having maximum degree . We show that every such graph has a strong edge-coloring using at most 3 colors. Our result confirms a conjecture of Brualdi and Quinn Massey (1993) for this class of bipartite graphs.

A red-blue graph is a graph where every edge is colored either red or blue. The exact perfect matching problem asks for a perfect matching in a red-blue graph that has exactly a given number of red edges. We show that for complete and bipartite complete graphs, the exact perfect matching problem is logspace equivalent to the perfect matching problem. Hence, an efficient parallel algorithm for perfect matching would carry over to the exact perfect matching problem for this class of graphs. We also report some progress in extending the result to arbitrary graphs.

In this article, we characterize various algebraic and order structures whose zero-divisor graphs are perfect graphs. We strengthen the result of Chenź(2003, Theorem 2.5) by providing a simpler proof of Beck's conjecture for the class of finite reduced rings (not necessarily commutative).

The Perfect Graph Theorems are important results in graph theory describing the relationship between clique number $\omega(G) $ and chromatic number $\chi(G) $ of a graph $G$. A graph $G$ is called \emph{perfect} if $\chi(H)=\omega(H)$ for every induced subgraph $H$ of $G$. The Strong Perfect Graph Theorem (SPGT) states that a graph is perfect if and only if it does not contain an odd hole (or an odd anti-hole) as its induced subgraph. The Weak Perfect Graph Theorem (WPGT) states that a graph is perfect if and only if its complement is perfect. In this paper, we present a formal framework for working with finite simple graphs. We model finite simple graphs in the Coq Proof Assistant by representing its vertices as a finite set over a countably infinite domain. We argue that this approach provides a formal framework in which it is convenient to work with different types of graph constructions (or expansions) involved in the proof of the Lov\'{a}sz Replication Lemma (LRL), which is also the key result used in the proof of Weak Perfect Graph Theorem. Finally, we use this setting to develop a constructive formalization of the Weak Perfect Graph Theorem., Comment: The 9th ACM SIGPLAN International Conference on Certified Programs and Proofs (CPP 2020)

Strongly perfect graphs have been studied by several authors (e.g. Berge and Duchet (1984) [1], Ravindra (1984) [12] and Wang (2006) [14]). In a series of two papers, the current paper being the first one, we investigate a fractional relaxation of strong perfection. Motivated by a wireless networking problem, we consider claw-free graphs that are fractionally strongly perfect in the complement. We obtain a forbidden induced subgraph characterization and display graph-theoretic properties of such graphs. It turns out that the forbidden induced subgraphs that characterize claw-free graphs that are fractionally strongly perfect in the complement are precisely the cycle of length 6, all cycles of length at least 8, four particular graphs, and a collection of graphs that are constructed by taking two graphs, each a copy of one of three particular graphs, and joining them in a certain way by a path of arbitrary length. Wang (2006) [14] gave a characterization of strongly perfect claw-free graphs. As a corollary of the results in this paper, we obtain a characterization of claw-free graphs whose complements are strongly perfect.

In 1962, Erd\H{o}s proved a theorem on the existence of Hamilton cycles in graphs with given minimum degree and number of edges. Significantly strengthening in case of balanced bipartite graphs, Moon and Moser proved a corresponding theorem in 1963. In this paper we establish several spectral analogues of Moon and Moser's theorem on Hamilton paths in balanced bipartite graphs and nearly balanced bipartite graphs. One main ingredient of our proofs is a structural result of its own interest, involving Hamilton paths in balanced bipartite graphs with given minimum degree and number of edges., Comment: 14 pages, 2 figures, Version 2 differs from Version 1 in improved presentation, to appear in Linear Algebra and Its Applications

We obtain a sharp result that for any even n ? 34, every {Dn,Dn+1}-regular graph of order n contains ?n/4? disjoint perfect matchings, where Dn = 2?n/4?-1. As a consequence, for any integer D ? Dn, every {D, D+1}- regular graph of order n contains (D-?n/4?+1) disjoint perfect matchings.

Let P t and C l denote a path on t vertices and a cycle on l vertices, respectively. In this paper we study the k-COLORING problem for (P t ,C l)-free graphs. It has been shown by Golovach, Paulusma, and Song that when l = 4 all these problems can be solved in polynomial time. By contrast, we show that in most other cases the k-COLORING problem for (P t ,C l)-free graphs is NP-complete. Specifically, for l = 5 we show that k-COLORING is NP-complete for (P t ,C 5)-free graphs when k ≥ 4 and t ≥ 7; for l ≥ 6 we show that k-COLORING is NP-complete for (P t ,C l)-free graphs when k ≥ 5, t ≥ 6; and additionally, we prove that 4-COLORING is NP-complete for (P t ,C l)-free graphs when t ≥ 7 and l ≥ 6 with l ≠ 7, and that 4-COLORING is NP-complete for (P t ,C l)-free graphs when t ≥ 9 and l ≥ 6 with l ≠ 9. It is known that, generally speaking, for large k the k-COLORING problem tends to remain NP-complete when one forbids an induced path P t with large t. Our findings mean that forbidding an additional induced cycle C l (with l > 4) does not help. We also revisit the problem of k-COLORING (P t ,C 4)-free graphs, in the case t = 6. (For t = 5 the k-COLORING problem is known to be polynomial even on just P 5-free graphs, for every k.) The algorithms of Golovach, Paulusma, and Song are not practical as they depend on Ramsey-type results, and end up using tree-decompositions with very high widths. We develop more practical algorithms for 3-COLORING and 4-COLORING on (P 6,C 4)-free graphs. Our algorithms run in linear time if a clique cutset decomposition of the input graph is given. Moreover, our algorithms are certifying algorithms. We provide a finite list of all minimal non-k-colorable (P 6,C 4)-free graphs, for k = 3 and k = 4. Our algorithms output one of these minimal obstructions whenever a k-coloring is not found. In fact, we prove that there are only finitely many minimal non-k-colorable (P 6,C 4)-free graphs for any fixed k; however, we do not have the explicit lists for higher k, and thus no certifying algorithms. (We note there are infinitely many non-k-colorable P 5-free, and hence P 6-free, graphs for any given k ≥ 4, according to a result of Hoang, Moore, Recoskie, Sawada, and Vatshelle.)

We present a new proof of an algebraic characterization of circle graphs due to W. Naji. For bipartite graphs, Naji's theorem is equivalent to an algebraic characterization of planar matroids due to J. Geelen and B. Gerards. Naji's theorem also yields an algebraic characterization of permutation graphs., v1: 22 pages, no figure. v2: minor improvements. 23 pages, no figure v3: minor improvements. 23 pages, no figure v4: substantial improvements reflecting reader's suggestions. 32 pages, no figure

We say that a graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal Hajnal-Szemeredi theorem 12 characterises the minimum degree that ensures a graph G contains a perfect K r -packing. Balogh, Kostochka and Treglown 4 proposed a degree sequence version of the Hajnal-Szemeredi theorem which, if true, gives a strengthening of the Hajnal-Szemeredi theorem. In this paper we prove this conjecture asymptotically. Another fundamental result in the area is the Alon-Yuster theorem 3 which gives a minimum degree condition that ensures a graph contains a perfect H-packing for an arbitrary graph H. We give a wide-reaching generalisation of this result by answering another conjecture of Balogh, Kostochka and Treglown 4 on the degree sequence of a graph that forces a perfect H-packing. We also prove a degree sequence result concerning perfect transitive tournament packings in directed graphs. The proofs blend together the regularity and absorbing methods.

Given two graphs \(H_1\) and \(H_2\), a graph is \((H_1,H_2)\)-free if it contains no induced subgraph isomorphic to \(H_1\) or \(H_2\). Let \(P_t\) and \(C_t\) be the path and the cycle on t vertices, respectively. A banner is the graph obtained from a \(C_4\) by adding a new vertex and making it adjacent to exactly one vertex of the \(C_4\). For a fixed integer \(k\ge 1\), a graph G is said to be k-vertex-critical if the chromatic number of G is k and the removal of any vertex results in a graph with chromatic number less than k. The study of k-vertex-critical graphs for graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there is a polynomial-time algorithm to decide if a graph in the class is \((k-1)\)-colorable. In this paper, we show that there are finitely many 6-vertex-critical (\(P_5\), banner)-free graphs. This is one of the few results on the finiteness of k-vertex-critical graphs when \(k>4\). To prove our result, we use the celebrated Strong Perfect Graph Theorem and well-known properties on k-vertex-critical graphs in a creative way.

Interaction between clique number \(\omega (G) \) and chromatic number \(\chi (G) \) of a graph is a well studied topic in graph theory. Perfect Graph Theorems are probably the most important results in this direction. Graph G is called perfect if \(\chi (H)=\omega (H)\) for every induced subgraph H of G. The Strong Perfect Graph Theorem (SPGT) states that a graph is perfect if and only if it does not contain an odd hole (or an odd anti-hole) as its induced subgraph. The Weak Perfect Graph Theorem (WPGT) states that a graph is perfect if and only if its complement is perfect. In this paper, we present a formal framework for verifying these results. We model finite simple graphs in the constructive type theory of Coq Proof Assistant without adding any axiom to it. Finally, we use this framework to present a constructive proof of the Lovasz Replication Lemma, which is the central idea in the proof of Weak Perfect Graph Theorem.

A trivial lower bound on the chromatic number of a graph is its clique number. In general, there is no upper bound on the chromatic number as a function of its clique number. Indeed, a famous result of Erdős establishes the existence of graphs with arbitrary large chromatic number and no triangle. However, well-structured classes of graphs may admit such a function: we say that a hereditary class of graphs, i.e., a class closed under taking induced subgraphs, is _$\chi$-bounded_ if there exists a function $f$ such that the chromatic number of every graph in the class is at most $f(\omega)$, where $\omega$ is its clique number. The most prominent $\chi$-bounded class of graphs is the class of [perfect graphs](https://en.wikipedia.org/wiki/Perfect_graph), i.e., graphs $G$ such that the chromatic number of every induced subgraph of $G$ is equal to its clique number. Here, the function $f$ from the definition of $\chi$-bounded class is the identity. The [Strong Perfect Graph Theorem](https://en.wikipedia.org/wiki/Strong_perfect_graph_theorem) of Chudnovsky, Robertson, Seymour and Thomas asserts that a graph is perfect if and only if it does not contain an odd cycle or its complement as an induced subgraph. Hence, perfect graphs include, among many other graph classes, interval graphs and more generally chordal graphs. The authors consider graphs with bounded rank-width, or equivalently [clique-width](https://en.wikipedia.org/wiki/Clique-width). Informally speaking, such graphs can be iteratively decomposed along simple structured cuts to single vertices. Decompositions along such cuts are assumed to play a key role in the characterization of graph classes closed under taking vertex minors analogously to the role of [tree decompositions](https://en.wikipedia.org/wiki/Tree_decomposition) in the [Graph Structure Theorem](https://en.wikipedia.org/wiki/Graph_structure_theorem) for graph classes closed under taking minors. Dvořak and Kraľ proved a conjecture of Geelen that every class of graphs with bounded rank-width is $\chi$-bounded, however, their function $f$ is exponential. The authors significantly improve this by showing that the function $f$ can be chosen to be polynomial and actually prove a more general statement concerning graphs decomposable along simple structured cuts to graphs from a polynomially $\chi$-bounded class. In the case of rank-width, the degree of the polynomial depends on the rank-width and this dependence is shown to be necessary.

The strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. This is an extension of the matching preclusion problem that was introduced by Park and Ihm. The generalized ( n , k ) -star graph was introduced to address scaling issues of the star graph, and it has many desirable properties. In this paper, the goal is to find the strong matching preclusion number of ( n , k ) -star graphs and to categorize all optimal strong matching preclusion sets of these graphs. Since bipartite graphs generally have low strong matching preclusion numbers, we assume that k ? n - 2 .

A connected undirected graph G is a Seymour graph if the maximum number of edge disjoint T -cuts is equal to the cardinality of a minimum T -join for every even subset T of V ( G ) . Ageev, Kostochka, and Szigeti characterized Seymour graphs in 1997. In this paper, we characterize minimal non-Seymour graphs. More precisely, we show that minimal non-Seymour graphs can be completely described by two infinite families of graphs, and we provide a procedure to construct them. Our characterization also generalizes a theorem of Lovasz concerning minimal nonbipartite matching-covered graphs.

In this paper we characterize all bipartite graphs with at most six non-zero eigenvalues. We determine the eigenvalues of bipartite graphs that have at most four nonzero eigenvalues. V tem članku karakteriziramo vse dvodelne grafe z največ šestimi neničelnimi lastnimi vrednostmi. Določimo lastne vrednosti dvodelnih grafov ki imajo največ štiri neničelne lastne vrednosti.

Several versions of the Riemann–Hurwitz theorem for branched coverings of graphs are presented. A larger class of graphs which may have not only multiple edges but also loops and darts is considered. This makes it possible to render the class of graphs closed with respect to morphisms arising as quotient maps for actions of finite groups.

Let G be a (molecule) graph. A perfect matching, or Kekule structure of G is a set of independent edges covering every vertex exactly once. Enumeration of Kekule structures of a graph is interest in chemistry, physics and mathematics. In this paper, we focus on the number of perfect matchings in polyomino chain graphs. Simple exact formulas are given for the expected value of the number of perfect matchings in random polyomino chain graphs and for the asymptotic behavior of this expectation. Moreover, the average value of the number of perfect matchings with respect to the set of all polyomino chain graphs with s square-cells.

The theory of matching energy of graphs since been proposed by Gutman and Wagner in 2012, has attracted more and more attention. It is denoted by B n , m , the class of bipartite graphs with order n and size m. In particular, B n , n denotes the set of bipartite unicyclic graphs, which is an interesting class of graphs. In this paper, for odd n, we characterize the bipartite unicyclic graphs with the first ? n - 3 4 ? largest matching energies. There is an interesting correspondence: we conclude that the graph with the second maximal matching energy in B n , n for odd n ? 11 is P n 6 , which is the only graph attaining the maximum value of the energy among all the (bipartite) unicyclic graphs for n ? 16.

Let G be a simple, undirected graph and Gw be the weighted graph obtained from G by giving weights to its edges using a positive weight function w. A weighted graph Gw is said to be nonsingular if its adjacency matrix A(Gw) is nonsingular. In [9], Godsil has given a class $\mathcal{G }$of connected, unweighted, bipartite, nonsingular graphs G with a unique perfect matching, such that A(G)−1 is signature similar to a nonnegative matrix, that is, there exists a diagonal matrix D with diagonal entries ±1 such that DA(G)−1D is nonnegative. The graph associated to the matrix DA(G)−1D is called the inverse of G and it is denoted by G+. The graph G+ is an undirected, weighted, connected, bipartite graph with a unique perfect matching. Nonsingular, unweighted trees are contained inside the class G. We first give a constructive characterization of the class of weighted graphs Hw that can occur as the inverse of some graph G∈\mathcal{ G}. This generalizes Theorem 2.6 of Neumann and Pati[13], where the authors have characterized graphs that occur as inverses of nonsingular, unweighted trees. A weighted graph Gw is said to have the property (R) if for each eigenvalue λ of A(Gw), 1⁄λ is also an eigenvalue of A(Gw). If further, the multiplicity of λ and 1⁄λ are the same, then Gw is said to have property (SR). A characterization of the class of nonsingular, weighted trees Tw with at least 8 vertices that have property (R) was given in [13] under some restriction on the weights. It is natural to ask for such a characterization for the whole of G, possibly with some weaker restrictions on the weights. We supply such a characterization. In particular, for trees it settles an open problem raised in [13].

Counting the number of perfect matchings in graphs is a computationally hard problem. However, in the case of planar graphs, and even for K3,3-free graphs, the number of perfect matchings can be computed efficiently. The technique to achieve this is to compute a Pfaffian orientation of a graph. In the case of K5-free graphs, this technique will not work because some K5-free graphs do not have a Pfaffian orientation. We circumvent this problem and show that the number of perfect matchings in K5-free graphs can be computed in polynomial time. We also parallelize the sequential algorithm and show that the problem is in TC2. We remark that our results generalize to graphs without singly-crossing minor.

We prove that if p ≫ lnn/n, then a.a.s. every subgraph of random bipartite graph G(n, n, p) with minimum degree at least (1/2 + o(l))np is Hamiltonian. The range of p and the constant 1/2 involved are both asymptotically best possible. The result can be viewed as a generalization of the Dirac theorem within the context of bipartite graphs. The proof uses Posa’s rotation and extension method and is closely related to a recent work of Lee and Sudakov.

A graph $G$ is \emph{trivially perfect} if for every induced subgraph the cardinality of the largest set of pairwise nonadjacent vertices (the stability number) $\alpha(G)$ equals the number of (maximal) cliques $m(G)$. We characterize the trivially perfect graphs in terms of vertex-coloring and we extend some definitions to infinite graphs.

Normal graphs are defined in terms of cross-intersecting set families: a graph is normal if it admits a clique cover $$\mathcal {Q}$$ and a stable set cover $$\mathcal {S}$$ s.t. every clique in $$\mathcal {Q}$$ intersects every stable set in $$\mathcal {S}$$ . Normal graphs can be considered as closure of perfect graphs by means of co-normal products and graph entropy. Perfect graphs have been characterized as those graphs without odd holes and odd antiholes as induced subgraphs (Strong Perfect Graph Theorem, Chudnovsky et al.). Korner and de Simone observed that $$C_5$$ , $$C_7$$ , and $$\overline{C}_7$$ are minimal not normal and conjectured, in analogy to the Strong Perfect Graph Theorem, that every $$(C_5, C_7, \overline{C}_7)$$ -free graph is normal (Normal Graph Conjecture, Korner and de Simone). Recently, this conjecture has been disproved by Harutyunyan, Pastor and Thomasse. However, in this paper we verify it for two classes of sparse graphs, 1-trees and cacti. In addition, we provide both linear time recognition algorithms and characterizations for the normal graphs within these two classes.

We prove that the strong perfect graph conjecture holds for graphs that do not contain parachutes or proper wheels. This is done by showing the following theorem: If a graph G contains no odd hole, no parachute and no proper wheel, then G is bipartite or the line graph of a bipartite graph or G contains a star cutset or an extended strong 2-join or Ḡ is disconnected. To prove this theorem, we prove two decomposition theorems which are interesting in their own rights. The first is a generalization of the Burlet–Fonlupt decomposition of Meyniel graphs by clique cutsets and amalgams. The second is a precursor of the recent decomposition theorem of Chudnovsky, Robertson, Seymour and Thomas for Berge graphs that contain a line graph of a bipartite subdivision of a 3-connected graph.

Since the critical graphs of order n ≤ 14 were studied so far [1] , [2] , [3] , [4] , [6] , [11] , in this paper, we are concerned with 5-critical graphs of order 15. This paper gives the complete possible list of degree sequences of 5-critical graphs of order 15.

We study the linear extension complexity of stable set polytopes of perfect graphs. We make use of known structural results permitting to decompose perfect graphs into basic perfect graphs by means of two graph operations: 2-join and skew partitions. Exploiting the link between extension complexity and the nonnegative rank of an associated slack matrix, we investigate the behaviour of the extension complexity under these graph operations. We show bounds for the extension complexity of the stable set polytope of a perfect graph $G$ depending linearly on the size of $G$ and involving the depth of a decomposition tree of $G$ in terms of basic perfect graphs., 17 pages

The mirror (or bipartite complement) $${{\mathrm{mir}}}(B)$$mir(B) of a bipartite graph $$B=(X,Y,E)$$B=(X,Y,E) has the same color classes $$X$$X and $$Y$$Y as $$B$$B, and two vertices $$x \in X$$x?X and $$y \in Y$$y?Y are adjacent in $${{\mathrm{mir}}}(B)$$mir(B) if and only if $$xy \notin E$$xy?E. A bipartite graph is chordal bipartite if none of its induced subgraphs is a chordless cycle with at least six vertices. In this paper, we deal with chordal bipartite graphs whose mirror is chordal bipartite as well; we call these graphs auto-chordal bipartite graphs (ACB graphs for short). We characterize ACB graphs, show that ACB graphs have unbounded bipartite Dilworth number, and we characterize ACB graphs with bipartite Dilworth number $$k$$k.

We characterize the graphs for which a linear relaxation of a facility location problem defines a polytope with all integral extreme points. We use a transformation to a stable set problem in perfect graphs. Based on this transformation, these graphs can be recognized in polynomial time.