Strong Perfect Graph Research Articles

Colouring graphs with no induced six-vertex path or diamond

The diamond is the graph obtained by removing an edge from the complete graph on 4 vertices. A graph is (P6, 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 (P6, 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 (P6, 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 (P6, diamond, K6)-free graph. Together with the Lovász theta function, this gives a polynomial time algorithm to compute the chromatic number of (P6, diamond)-free graphs.

Forbidden Induced Pairs for Perfectness and $\omega$-Colourability of Graphs

We characterise the pairs of graphs $\{ X, Y \}$ such that all $\{ X, Y \}$-free graphs (distinct from $C_5$) are perfect. Similarly, we characterise pairs $\{ X, Y \}$ such that all $\{ X, Y \}$-free graphs (distinct from $C_5$) are $\omega$-colourable (that is, their chromatic number is equal to their clique number). More generally, we show characterizations of pairs $\{ X, Y \}$ for perfectness and $\omega$-colourability of all connected $\{ X, Y \}$-free graphs which are of independence at least~$3$, distinct from an odd cycle, and of order at least $n_0$, and similar characterisations subject to each subset of these additional constraints. (The classes are non-hereditary and the characterisations for perfectness and $\omega$-colourability are different.) We build on recent results of Brause et al. on $\{ K_{1,3}, Y \}$-free graphs, and we use Ramsey's Theorem and the Strong Perfect Graph Theorem as main tools. We relate the present characterisations to known results on forbidden pairs for $\chi$-boundedness and deciding $k$-colourability in polynomial time.

On chromatic number and perfectness of fuzzy graph

Perfect graphs have a significant role in graph theory in terms of structure as well as application. This article embarked a comprehensive study on perfect fuzzy graphs attempted to obtain the fuzzy analog of the famous weak and strong perfect graph theorems following due classification of fuzzy graphs in terms of perfectness. The inter-relationship between chromatic numbers of a fuzzy graph and its complement graph has been studied in detail deriving important theories and results. It has been shown that unlike crisp case, for fuzzy graphs there exists one and only one perfect fuzzy graph theorem satisfying both the conditions of strong and weak perfect graph theorems. A detailed study of the perfectness of different types of fuzzy graphs has been carried out comparing the results with those obtained on crisp graphs. It has been shown through illustrative examples how the perfectness of a fuzzy graph and that of its underlying crisp graph is independent of each other. Finally, the use of perfect fuzzy graph on application grounds has been evaluated by using it in analyzing how a huge percentage of population in a city is unknowingly living under a big threat due exposure to electromagnetic radiation emitted from cell-phone towers.

Point Partition Numbers: Perfect Graphs

Graphs considered in this paper are finite, undirected and without loops, but with multiple edges. For an integer $$t\ge 1$$ , denote by $$\mathcal{MG}_t$$ the class of graphs whose maximum multiplicity is at most t. A graph G is called strictly t-degenerate if every non-empty subgraph H of G contains a vertex v whose degree in H is at most $$t-1$$ . The point partition number $$\chi _t(G)$$ of G is the smallest number of colors needed to color the vertices of G so that each vertex receives a color and vertices with the same color induce a strictly t-degenerate subgraph of G. So $$\chi _1$$ is the chromatic number, and $$\chi _2$$ is known as the point aboricity. The point partition number $$\chi _t$$ with $$t\ge 1$$ was introduced by Lick and White (Can J Math 22:1082–1096, 1970). If H is a simple graph, then tH denotes the graph obtained from H by replacing each edge of H by t parallel edges. Then $$\omega _t(G)$$ is the largest integer n such that G contains a $$tK_n$$ as a subgraph. Let G be a graph belonging to $$\mathcal{MG}_t$$ . Then $$\omega _t(G)\le \chi _t(G)$$ and we say that G is $$\chi _t$$ -perfect if every induced subgraph H of G satisfies $$\omega _t(H)=\chi _t(H)$$ . Based on the Strong Perfect Graph Theorem due to Chudnowsky, Robertson, Seymour and Thomas (Ann Math 164:51–229, 2006), we give a characterization of $$\chi _t$$ -perfect graphs of $$\mathcal{MG}_t$$ by a set of forbidden induced subgraphs (see Theorems 2 and 3). We also discuss some complexity problems for the class of $$\chi _t$$ -critical graphs.

An optimal χ‐bound for (P6, diamond)‐free graphs

AbstractIn this paper we show that every (, diamond)‐free graph satisfies , where and are the chromatic number and clique number of , respectively. Our bound is attained by the complement of the famous 27‐vertex Schläfli graph. Our result unifies previously known results on the existence of linear ‐binding functions for several graph classes. Our proof is based on a reduction via the Strong Perfect Graph Theorem to imperfect (, diamond)‐free graphs, a careful analysis of the structure of those graphs, and a computer search that relies on a well‐known characterization of 3‐colourable ‐free graphs.

The Normal Graph Conjecture for Two Classes of Sparse 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.

Polyhedral results on the stable set problem in graphs containing even or odd pairs

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 short proof of the wonderful lemma

AbstractThe Wonderful Lemma, that was first proved by Roussel and Rubio, is one of the most important tools in the proof of the Strong Perfect Graph Theorem. Here we give a short proof of this lemma.

Counting Homomorphisms to Square-Free Graphs, Modulo 2

We study the problem ⊕H oms T o H of counting, modulo 2, the homomorphisms from an input graph to a fixed undirected graph H . A characteristic feature of modular counting is that cancellations make wider classes of instances tractable than is the case for exact (nonmodular) counting; thus, subtle dichotomy theorems can arise. We show the following dichotomy: for any H that contains no 4-cycles, ⊕H oms T o H is either in polynomial time or is ⊕P-complete. This partially confirms a conjecture of Faben and Jerrum that was previously only known to hold for trees and for a restricted class of tree-width-2 graphs called cactus graphs. We confirm the conjecture for a rich class of graphs, including graphs of unbounded tree-width. In particular, we focus on square-free graphs, which are graphs without 4-cycles. These graphs arise frequently in combinatorics, for example, in connection with the strong perfect graph theorem and in certain graph algorithms. Previous dichotomy theorems required the graph to be tree-like so that tree-like decompositions could be exploited in the proof. We prove the conjecture for a much richer class of graphs by adopting a much more general approach.

Clique–Stable Set separation in perfect graphs with no balanced skew-partitions

Inspired by a question of Yannakakis on the Vertex Packing polytope of perfect graphs, we study the Clique–Stable Set separation in a non-hereditary subclass of perfect graphs. A cut (B,W) of G (a bipartition of V(G)) separates a clique K and a stable set S if K⊆B and S⊆W. A Clique–Stable Set separator is a family of cuts such that for every clique K, and for every stable set S disjoint from K, there exists a cut in the family that separates K and S. Given a class of graphs, the question is to know whether every graph of the class admits a Clique–Stable Set separator containing only polynomially many cuts. It was recently proved to be false for the class of all graphs (Göös, 2015), but it remains open for perfect graphs, which was Yannakakis’ original question. Here we investigate this problem on perfect graphs with no balanced skew-partition; the balanced skew-partition was introduced in the decomposition theorem of Berge graphs which led to the celebrated proof of the Strong Perfect Graph Theorem. Recently, Chudnovsky, Trotignon, Trunck and Vušković proved that forbidding this unfriendly decomposition permits to recursively decompose Berge graphs (more precisely, Berge trigraphs) using 2-join and complement 2-join until reaching a “basic” graph, and in this way, they found an efficient combinatorial algorithm to color those graphs.We apply their decomposition result to prove that perfect graphs with no balanced skew-partition admit a quadratic-size Clique–Stable Set separator, by taking advantage of the good behavior of 2-join with respect to this property. We then generalize this result and prove that the Strong Erdős–Hajnal property holds in this class, which means that every such graph has a linear-size biclique or complement biclique. This is remarkable since the property does not hold for all perfect graphs (Fox, 2006), and this is motivated here by the following statement: when the Strong Erdős–Hajnal property holds in a hereditary class of graphs, then both the Erdős–Hajnal property and the polynomial Clique–Stable Set separation hold. Finally, we define the generalized k-join and generalize both our results on classes of graphs admitting such a decomposition.

Applications of Super Strongly Perfect Graph for Manufacturing System towards a Leaner Structure

Some restructuring decisions are conceptualized which reflect the aim of the organization to gradually evolve the manufacturing system towards a leaner structure. This is done by way of defining simplified process so that lesser hindrance in terms of cycles of interactions is found. The reframing decisions are given by five restructured configurations of the manufacturing system. Models using graph theory are developed for original configuration and each of the new reframed configurations and the resulting structural characterization information is used to compare the structure of restructured configurations with the original configuration. A graph G is Super Strongly Perfect (SSP) if every induced sub graph H of G possesses a minimal dominating set that meets all the maximal cliques of H. A study on some classes of super strongly perfect graphs like wheel and double wheel graphs (in which each graph represents structure of some manufacturing system) are given.

Perfect Graphs with No Balanced Skew‐Partition are 2‐Clique‐Colorable

AbstractA graph G is perfect if for all induced subgraphs H of G, . A graph G is Berge if neither G nor its complement contains an induced odd cycle of length at least five. The Strong Perfect Graph Theorem [9] states that a graph is perfect if and only if it is Berge. The Strong Perfect Graph Theorem was obtained as a consequence of a decomposition theorem for Berge graphs [M. Chudnovsky, Berge trigraphs and their applications, PhD thesis, Princeton University, 2003; M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, Ann Math 164 (2006), 51–229.], and one of the decompositions in this decomposition theorem was the “balanced skew‐partition.” A clique‐coloring of a graph G is an assignment of colors to the vertices of G in such a way that no inclusion‐wise maximal clique of G of size at least two is monochromatic, and the clique‐chromatic number of G, denoted by , is the smallest number of colors needed to clique‐color G. There exist graphs of arbitrarily large clique‐chromatic number, but it is not known whether the clique‐chromatic number of perfect graphs is bounded. In this article, we prove that every perfect graph that does not admit a balanced skew‐partition is 2‐clique colorable. The main tool used in the proof is a decomposition theorem for “tame Berge trigraphs” due to Chudnovsky et al. (http://arxiv.org/abs/1308.6444).

Counting List Matrix Partitions of Graphs

Given a symmetric $D\times D$ matrix $M$ over \0,1,*\, a list $M$-partition of a graph $G$ is a partition of the vertices of $G$ into $D$ parts which are associated with the rows of $M$. The part of each vertex is chosen from a given list in such a way that no edge of $G$ is mapped to a 0 in $M$ and no nonedge of $G$ is mapped to a 1 in $M$. Many important graph-theoretic structures can be represented as list $M$-partitions including graph colorings, split graphs, and homogeneous sets and pairs, which arise in the proofs of the weak and strong perfect graph conjectures. Thus, there has been quite a bit of work on determining for which matrices $M$ computations involving list $M$-partitions are tractable. This paper focuses on the problem of counting list $M$-partitions, given a graph $G$ and given a list for each vertex of $G$. We identify a certain set of “tractable” matrices $M$. We give an algorithm that counts list $M$-partitions in polynomial time for every (fixed) matrix $M$ in this set. The algorit...

Super Strongly Perfect Graphs in the Selection of Cluster Heads in Wireless Sensor Networks

Wireless sensor network comprises of dense sensor nodes which are randomly deployed. Major challenges in WSN are limited battery source and computation capacity. Considerable research has been carried out in the area of maximizing battery lifetime by reducing the energy consumption. Once such proposed technique involves hierarchical topology control. Conventionally proposed algorithm for hierarchical topology control involves computationally intensive soft computing tools. It leads to higher energy consumption in the sink node. Hence it necessitates computationally less intensive technique for cluster head selection. In this paper, an efficient cluster head selection is proposed using minimal dominating set in Super Strongly Perfect (SSP) graph.

Perfect Digraphs

AbstractThe clique number of a digraph D is the size of the largest bidirectionally complete subdigraph of D. D is perfect if, for any induced subdigraph H of D, the dichromatic number defined by Neumann‐Lara (The dichromatic number of a digraph, J. Combin. Theory Ser. B 33 (1982), 265–270) equals the clique number . Using the Strong Perfect Graph Theorem (M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, Ann. Math. 164 (2006), 51–229) we give a characterization of perfect digraphs by a set of forbidden induced subdigraphs. Modifying a recent proof of Bang‐Jensen et al. (Finding an induced subdivision of a digraph, Theoret. Comput. Sci. 443 (2012), 10–24) we show that the recognition of perfect digraphs is co‐‐complete. It turns out that perfect digraphs are exactly the complements of clique‐acyclic superorientations of perfect graphs. Thus, we obtain as a corollary that complements of perfect digraphs have a kernel, using a result of Boros and Gurvich (Perfect graphs are kernel solvable, Discrete Math. 159 (1996), 35–55). Finally, we prove that it is ‐complete to decide whether a perfect digraph has a kernel.

Envy-free pricing in multi-item markets

In this article, we study revenue maximizing envy-free pricing in multi-item markets: There are m indivisible items with unit supply each and n potential buyers where each buyer is interested in acquiring one item. The goal is to determine allocations (a matching between buyers and items) and prices of all items to maximize total revenue given that all buyers are envy-free. We give a polynomial time algorithm to compute a revenue maximizing envy-free pricing when every buyer evaluates at most two items at a positive valuation, by reducing it to an instance of weighted independent set in a perfect graph and applying the Strong Perfect Graph Theorem. We complement our result by showing that the problem becomes NP-hard if some buyers are interested in at least three items.

Fast recognition of doubled graphs

Double split graphs form one of the five basic classes in the proof of the strong perfect graph theorem by Chudnovsky et al. [3]. A doubled graph is any induced subgraph of a double split graph. We show here that deciding if a graph G is a doubled graph can be done in time O(|V(G)|+|E(G)|) by analyzing the degree structure of the graph.

SUPER STRONGLY PERFECTNESS OF SOME GRAPHS

A Graph G is Super Strongly Perfect Graph if every induced sub graph H of G possesses a minimal dominating set that meets all the maximal cliques of H. The structure of Super Strongly Perfect Graphs have been characterized by some classes of graphs like Cycle graphs, Circulant graphs, Complete graphs, Complete Bipartite graphs etc., In this paper, we have analysed some other graph classes like, Bicyclic graphs, Dumb bell graphs and Star graphs to characterize the structure of Super Strongly Perfect Graphs in a different way. By this we found the cardinality of minimal dominating set and maximal cliques of the above graphs. AMS Subject Classification: 05C75

An investigation on some classes of super strongly perfect graphs

A Graph G is Super Strongly Perfect Graph if every induced sub graph H of G possesses a minimal dominating set that meets all the maximal complete sub graphs of H. Bipartite graphs, Complete graphs etc., are some of the most important classes of Super Strongly Perfect graphs. Here, we summarize the results concerning Super Strongly Perfect graphs. We investigate some classes of Super Strongly Perfect graphs and we investigate the structure of Super Strongly Perfect Graphs.

Packing and covering with linear programming: A survey

This paper considers the polyhedral results and the min–max results on packing and covering problems of the decade. Since the strong perfect graph theorem (published in 2006), the main such results are available for the packing problem, however there are still important polyhedral questions that remain open. For the covering problem, the main questions are still open, although there has been important progress. We survey some of the main results with emphasis on those where linear programming and graph theory come together. They mainly concern the covering of cycles or dicycles in graphs or signed graphs, either with vertices or edges; this includes the multicut and integral multiflow problems.