Class Of Perfect Graphs Research Articles

Chromatic choosability for some classes of perfect graphs

A graph [Formula: see text] is called [Formula: see text]-list colorable if there is a vertex coloring of [Formula: see text] in which the color assigned to a vertex [Formula: see text] is chosen from a list [Formula: see text] associated with this vertex. We say [Formula: see text] is [Formula: see text]-choosable if all lists [Formula: see text] have the cardinality [Formula: see text] and [Formula: see text] is [Formula: see text]-list colorable for all possible assignments of such lists. A graph [Formula: see text] is said to be chromatic choosable if its chromatic and list chromatic numbers are equal. Investigation of [Formula: see text]-choosable graphs is one of the open problems. In this paper, we investigate this problem for classes of perfect graphs.

Adding a Tail in Classes of Perfect Graphs

Consider a graph G which belongs to a graph class C. We are interested in connecting a node w∉V(G) to G by a single edge uw where u∈V(G); we call such an edge a tail. As the graph resulting from G after the addition of the tail, denoted G+uw, need not belong to the class C, we want to compute the number of non-edges of G in a minimum C-completion of G+uw, i.e., the minimum number of non-edges (excluding the tail uw) to be added to G+uw so that the resulting graph belongs to C. In this paper, we study this problem for the classes of split, quasi-threshold, threshold and P4-sparse graphs and we present linear-time algorithms by exploiting the structure of split graphs and the tree representation of quasi-threshold, threshold and P4-sparse graphs.

On finding separators in temporal split and permutation graphs

We study the NP-hard Temporal(s,z)-Separation problem on temporal graphs, which are graphs with fixed vertex sets but edge sets that change over discrete time steps. We tackle Temporal(s,z)-Separation by restricting the layers (i.e., graphs characterized by edges that are present at a certain point in time) to specific graph classes. We restrict the layers of the temporal graphs to be either all split graphs or all permutation graphs (both being perfect graph classes) and provide both intractability and tractability results. In particular, we show that in general Temporal(s,z)-Separation remains NP-hard both on temporal split and temporal permutation graphs, but we also spot promising islands of fixed-parameter tractability particularly based on parameterizations that measure the amount of “change over time”.

Defective Coloring on Classes of Perfect Graphs

In Defective Coloring we are given a graph $G$ and two integers $\chi_d$, $\Delta^*$ and are asked if we can $\chi_d$-color $G$ so that the maximum degree induced by any color class is at most $\Delta^*$. We show that this natural generalization of Coloring is much harder on several basic graph classes. In particular, we show that it is NP-hard on split graphs, even when one of the two parameters $\chi_d$, $\Delta^*$ is set to the smallest possible fixed value that does not trivialize the problem ($\chi_d = 2$ or $\Delta^* = 1$). Together with a simple treewidth-based DP algorithm this completely determines the complexity of the problem also on chordal graphs. We then consider the case of cographs and show that, somewhat surprisingly, Defective Coloring turns out to be one of the few natural problems which are NP-hard on this class. We complement this negative result by showing that Defective Coloring is in P for cographs if either $\chi_d$ or $\Delta^*$ is fixed; that it is in P for trivially perfect graphs; and that it admits a sub-exponential time algorithm for cographs when both $\chi_d$ and $\Delta^*$ are unbounded.

OPEN PACKING NUMBER FOR SOME CLASSES OF PERFECT GRAPHS

Let \(G\) be a graph with the vertex set \(V(G)\). A subset \(S\) of \(V(G)\) is an open packing set of \(G\) if every pair of vertices in \(S\) has no common neighbor in \(G.\) The maximum cardinality of an open packing set of \(G\) is the open packing number of \(G\) and it is denoted by \(\rho^o(G)\). In this paper, the exact values of the open packing numbers for some classes of perfect graphs, such as split graphs, \(\{P_4, C_4\}\)-free graphs, the complement of a bipartite graph, the trestled graph of a perfect graph are obtained.

Mind the independence gap

The independence gap of a graph was introduced by Ekim et al. in 2018 as a measure of how far a graph is from being well-covered. It is defined as the difference between the maximum and minimum size of a maximal independent set.We investigate the independence gap of a graph from structural and algorithmic points of view, with a focus on classes of perfect graphs. Generalizing results on well-covered graphs due to Dean and Zito (1994) and Hujdurović et al. (2018), we express the independence gap of a perfect graph in terms of clique partitions and use this characterization to develop a polynomial-time algorithm for recognizing graphs of constant independence gap in any class of perfect graphs of bounded clique number. Next, we introduce a hereditary variant of the parameter, which we call hereditary independence gap and which measures the maximum independence gap over all induced subgraphs of the graph. We show that determining whether a given graph has hereditary independence gap at most k is polynomial-time solvable if k is fixed and co-NP-complete if k is part of input. We also investigate the complexity of the independent set and independent domination problems in graph classes related to independence gap. In particular, we show that the two problems are NP-complete in the class of graphs of independence gap at most one and that the weighted independent set problem is polynomial-time solvable in any class of graphs with bounded hereditary independence gap. Combined with some known results on claw-free graphs, our results imply that the independent domination problem is solvable in polynomial time in the class of {claw, 2P3}-free graphs.

Small 1-defective Ramsey numbers in perfect graphs

In this paper, we initiate the study of defective Ramsey numbers for the class of perfect graphs. Let PG be the class of all perfect graphs and R1PG(i,j) denote the smallest n such that all perfect graphs on n vertices have either a 1-dense i-set or a 1-sparse j-set. We show that R1PG(3,j)=j for any j≥2, R1PG(4,4)=6, R1PG(4,5)=8, R1PG(4,6)=10, R1PG(4,7)=13, R1PG(4,8)=15 and R1PG(5,5)=13. We exhibit all extremal graphs for R1PG(4,7)=13 (there are exactly three). We also obtain the 1-defective Ramsey number of order (4,7) in triangle-free perfect graphs, namely R1ΔPG(4,7)=12.

Clique‐cutsets beyond chordal graphs

AbstractTruemper configurations (thetas, pyramids, prisms, and wheels) have played an important role in the study of complex hereditary graph classes (eg, the class of perfect graphs and the class of even‐hole‐free graphs), appearing both as excluded configurations, and as configurations around which graphs can be decomposed. In this paper, we study the structure of graphs that contain (as induced subgraphs) no Truemper configurations other than (possibly) universal wheels and twin wheels. We also study several subclasses of this class. We use our structural results to analyze the complexity of the recognition, maximum weight clique, maximum weight stable set, and optimal vertex coloring problems for these classes. Furthermore, we obtain polynomial ‐bounding functions for these classes.

Reconfiguration of colorable sets in classes of perfect graphs

A set of vertices in a graph is c-colorable if the subgraph induced by the set has a proper c-coloring. In this paper, we study the problem of finding a step-by-step transformation (called a reconfiguration sequence) between two c-colorable sets in the same graph. This problem generalizes the well-studied Independent Set Reconfiguration problem. As the first step toward a systematic understanding of the complexity of this general problem, we study the problem on classes of perfect graphs. We first focus on interval graphs and give a combinatorial characterization of the distance between two c-colorable sets. This gives a linear-time algorithm for finding an actual shortest reconfiguration sequence for interval graphs. Since interval graphs are exactly the graphs that are simultaneously chordal and co-comparability, we then complement the positive result by showing that even deciding reachability is PSPACE-complete for chordal graphs and for co-comparability graphs. The hardness for chordal graphs holds even for split graphs. We also consider the case where c is a fixed constant and show that in such a case the reachability problem is polynomial-time solvable for split graphs but still PSPACE-complete for co-comparability graphs. The complexity of this case for chordal graphs remains unsettled. As by-products, our positive results give the first polynomial-time solvable cases (split graphs and interval graphs) for Feedback Vertex Set Reconfiguration.

Perfect graphs involving semitotal and semipaired domination

Let G be a graph with vertex set V and no isolated vertices, and let S be a dominating set of V. The set S is a semitotal dominating set of G if every vertex in S is within distance 2 of another vertex of S. And, S is a semipaired dominating set of G if S can be partitioned into 2-element subsets such that the vertices in each 2-set are at most distance two apart. The semitotal domination number $$\gamma _\mathrm{t2}(G)$$ is the minimum cardinality of a semitotal dominating set of G, and the semipaired domination number $$\gamma _\mathrm{pr2}(G)$$ is the minimum cardinality of a semipaired dominating set of G. For a graph without isolated vertices, the domination number $$\gamma (G)$$ , the total domination $$\gamma _t(G)$$ , and the paired domination number $$\gamma _\mathrm{pr}(G)$$ are related to the semitotal and semipaired domination numbers by the following inequalities: $$\gamma (G) \le \gamma _\mathrm{t2}(G) \le \gamma _t(G) \le \gamma _\mathrm{pr}(G)$$ and $$\gamma (G) \le \gamma _\mathrm{t2}(G) \le \gamma _\mathrm{pr2}(G) \le \gamma _\mathrm{pr}(G) \le 2\gamma (G)$$ . Given two graph parameters $$\mu $$ and $$\psi $$ related by a simple inequality $$\mu (G) \le \psi (G)$$ for every graph G having no isolated vertices, a graph is $$(\mu ,\psi )$$ -perfect if every induced subgraph H with no isolated vertices satisfies $$\mu (H) = \psi (H)$$ . Alvarado et al. (Discrete Math 338:1424–1431, 2015) consider classes of $$(\mu ,\psi )$$ -perfect graphs, where $$\mu $$ and $$\psi $$ are domination parameters including $$\gamma $$ , $$\gamma _t$$ and $$\gamma _\mathrm{pr}$$ . We study classes of perfect graphs for the possible combinations of parameters in the inequalities when $$\gamma _\mathrm{t2}$$ and $$\gamma _\mathrm{pr2}$$ are included in the mix. Our results are characterizations of several such classes in terms of their minimal forbidden induced subgraphs.

Graphs vertex-partitionable into strong cliques

A clique in a graph is strong if it intersects all maximal independent sets. A graph is localizable if it has a partition of the vertex set into strong cliques. Localizable graphs were introduced by Yamashita and Kameda in 1999 and form a rich class of well-covered graphs that coincides with the class of well-covered graphs within the class of perfect graphs. In this paper, we give several equivalent formulations of the property of localizability and develop polynomially testable characterizations of localizable graphs within three non-perfect graph classes: triangle-free graphs, C4-free graphs, and line graphs. Furthermore, we use localizable graphs to construct an infinite family of counterexamples to a conjecture due to Zaare-Nahandi about k-partite well-covered graphs having all maximal cliques of size k.

Star Coloring of Certain Graph Classes

A star coloring of an undirected graph G is a coloring of the vertices of G such that (i) no two adjacent vertices receive the same color, and (ii) no path on four vertices (not necessarily induced) is bi-colored. The star chromatic number of G, denoted by $$\chi _s(G)$$ , is the minimum number of colors needed to star color G. Similar to the notion of $$\chi $$ -boundedness in graphs, we say that a hereditary class of graphs $$\mathcal {G}$$ is $$\chi _s$$ -bounded if, for some integer valued function f, $$\chi _s(G) \le f(\omega (G))$$ for every $$G \in \mathcal {G}$$ , where $$\omega (G)$$ is the maximum number of vertices that are pairwise adjacent in G. We show that some classes of perfect graphs, some classes of ( $$P_5, C_4$$ )-free graphs, and some classes of $$K_{1, 3}$$ -free graphs are $$\chi _s$$ -bounded. We also give examples in most of the cases to show that the bounds are tight.

Clique cutsets beyond chordal graphs

Truemper configurations (thetas, pyramids, prisms, and wheels) have played an important role in the study of complex hereditary graph classes (e.g. the class of perfect graphs and the class of even-hole-free graphs), appearing both as excluded configurations, and as configurations around which graphs can be decomposed. In this paper, we study the structure of graphs that contain (as induced subgraphs) no Truemper configurations other than (possibly) universal wheels and twin wheels. We also study several subclasses of this class. We use our structural results to analyze the complexity of the recognition, maximum weight clique, maximum weight stable set, and optimal vertex coloring problems for these classes. We also obtain polynomial χ-bounding functions for these classes.

Chromatic and clique numbers of a class of perfect graphs

Let p be a prime number and n be a positive integer. The graph Gp(n) is a graph with vertex set [n] = f 1;2;:::;ng , in which there is an arc from u to v if and only if u v and p ∤ u+v. In this paper

Suppression distance computation for hierarchical clusterings

We discuss the computation of a distance between two hierarchical clusterings of the same set. It is defined as the minimum number of elements that have to be removed so the remaining clusterings are equal. The problem of distance computing was extensively studied for partitions. We prove it can be solved in polynomial time in the case of hierarchies as it gives birth to a class of perfect graphs. We also propose an algorithm based on recursively computing maximum assignments.

Polynomial-time algorithms for Subgraph Isomorphism in small graph classes of perfect graphs

Given two graphs, Subgraph Isomorphism is the problem of deciding whether the first graph (the base graph) contains a subgraph isomorphic to the second one (the pattern graph). This problem is NP-complete even for very restricted graph classes such as connected proper interval graphs. Only a few cases are known to be polynomial-time solvable even if we restrict the graphs to be perfect. For example, if both graphs are co-chain graphs, then the problem can be solved in linear time.In this paper, we present a polynomial-time algorithm for the case where the base graphs are chordal graphs and the pattern graphs are co-chain graphs. We also present a linear-time algorithm for the case where the base graphs are trivially perfect graphs and the pattern graphs are threshold graphs. These results answer some of the open questions of Kijima et al. (2012). To present a complexity contrast, we then show that even if the base graphs are somewhat restricted perfect graphs, the problem of finding a pattern graph that is a chain graph, a co-chain graph, or a threshold graph is NP-complete.

Graph classes and Ramsey numbers

For a graph class G and any two positive integers i and j, the Ramsey number RG(i,j) is the smallest positive integer such that every graph in G on at least RG(i,j) vertices has a clique of size i or an independent set of size j. For the class of all graphs, Ramsey numbers are notoriously hard to determine, and they are known only for very small values of i and j. Even if we restrict G to be the class of claw-free graphs, it is highly unlikely that a formula for determining RG(i,j) for all values of i and j will ever be found, as there are infinitely many nontrivial Ramsey numbers for claw-free graphs that are as difficult to determine as for arbitrary graphs. Motivated by this difficulty, we establish here exact formulas for all Ramsey numbers for three important subclasses of claw-free graphs: line graphs, long circular interval graphs, and fuzzy circular interval graphs. On the way to obtaining these results, we also establish all Ramsey numbers for the class of perfect graphs. Such positive results for graph classes are rare: a formula for determining RG(i,j) for all values of i and j, when G is the class of planar graphs, was obtained by Steinberg and Tovey (1993), and this seems to be the only previously known result of this kind. We complement our aforementioned results by giving exact formulas for determining all Ramsey numbers for several graph classes related to perfect graphs.

On the complexity of the black-and-white coloring problem on some classes of perfect graphs

Given a graph G and integers b and w. The black-and-white coloring problem asks if there exist disjoint sets of vertices B and W with |B|=b and |W|=w such that no vertex in B is adjacent to any vertex in W. In this paper we show that the problem is polynomial when restricted to cographs, interval graphs, permutation graphs, distance-hereditary graphs, and strongly chordal graphs. We show that the problem is NP-complete on splitgraphs.

An Intersection Model for Multitolerance Graphs: Efficient Algorithms and Hierarchy

Tolerance graphs model interval relations in such a way that intervals can tolerate a certain degree of overlap without being in conflict. This class of graphs has attracted many research efforts, mainly due to its interesting structure and its numerous applications, especially in DNA sequence analysis and resource allocation, among others. In one of the most natural generalizations of tolerance graphs, namely multitolerance graphs, two tolerances are allowed for each interval—one from the left and one from the right side of the interval. Then, in its interior part, every interval tolerates the intersection with others by an amount that is a convex combination of its two border-tolerances. In the comparison of DNA sequences between different organisms, the natural interpretation of this model lies on the fact that, in some applications, we may want to treat several parts of the genomic sequences differently. That is, we may want to be more tolerant at some parts of the sequences than at others. These two tolerances for every interval—together with their convex hull—define an infinite number of the so called tolerance-intervals, which make the multitolerance model inconvenient to cope with. In this article we introduce the first non-trivial intersection model for multitolerance graphs, given by objects in the 3-dimensional space called trapezoepipeds. Apart from being important on its own, this new intersection model proves to be a powerful tool for designing efficient algorithms. Given a multitolerance graph with n vertices and m edges along with a multitolerance representation, we present algorithms that compute a minimum coloring and a maximum clique in optimal O(nlogn) time, and a maximum weight independent set in O(m+nlogn) time. Moreover, our results imply an optimal O(nlogn) time algorithm for the maximum weight independent set problem on tolerance graphs, thus closing the complexity gap for this problem. Additionally, by exploiting more the new 3D-intersection model, we completely classify multitolerance graphs in the hierarchy of perfect graphs. The resulting hierarchy of classes of perfect graphs is complete, i.e. all inclusions are strict.

Indicated coloring of graphs

We study a graph coloring game in which two players collectively color the vertices of a graph in the following way. In each round the first player (Ann) selects a vertex, and then the second player (Ben) colors it properly, using a fixed set of colors. The goal of Ann is to achieve a proper coloring of the whole graph, while Ben is trying to prevent realization of this project. The smallest number of colors necessary for Ann to win the game on a graph G (regardless of Ben’s strategy) is called the indicated chromatic number of G, and denoted by χi(G). We approach the question how much χi(G) differs from the usual chromatic number χ(G). In particular, whether there is a function f such that χi(G)⩽f(χ(G)) for every graph G. We prove that f cannot be linear with leading coefficient less than 4/3. On the other hand, we show that the indicated chromatic number of random graphs is bounded roughly by 4χ(G). We also exhibit several classes of graphs for which χi(G)=χ(G) and show that this equality for any class of perfect graphs implies Clique-Pair Conjecture for this class of graphs.