Class Of Chordal Graphs Research Articles

Minimal toughness in special graph classes

Let $t$ be a positive real number. A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components, and all graphs are considered 0-tough. The toughness of a graph is the largest $t$ for which the graph is $t$-tough, whereby the toughness of complete graphs is defined as infinity. A graph is minimally $t$-tough if the toughness of the graph is $t$, and the deletion of any edge from the graph decreases the toughness. In this paper, we investigate the minimum degree and the recognizability of minimally $t$-tough graphs in the classes of chordal graphs, split graphs, claw-free graphs, and $2K_2$-free graphs.

All longest cycles in a 2‐connected partial 3‐tree share a common vertex

AbstractIn 1966 Gallai asked whether all longest paths in a connected graph have nonempty intersection. This is not true in general and various counterexamples have been found. In this paper, we study the related question on longest cycles instead of longest paths (to make the question valid, we require 2‐connectivity). The answer for cycles is also negative and not so much attention has been given to it. Classes for which the answer is known to be positive are dually chordal graphs and 3‐trees. Perhaps the main open class for the question of both paths and cycles is the class of chordal graphs. In this paper, we show that all longest cycles intersect in graphs with treewidth at most three. This class of graphs includes several important subclasses of graphs, such as series‐parallel graphs and ‐free chordal graphs.

Wait-free approximate agreement on graphs

Approximate agreement is one of the few variants of consensus that can be solved in a wait-free manner in asynchronous systems where processes communicate by reading and writing to shared memory. In this work, we consider a natural generalisation of approximate agreement on arbitrary undirected connected graphs. Each process is given a node of the graph as input and, if non-faulty, must output a node such that–all the outputs are within distance 1 of one another, and–each output value lies on a shortest path between two input values. From prior work, it is known that there is no wait-free algorithm among n≥3 processes for this problem on any cycle of length c≥4, by reduction from 2-set agreement (Castañeda et al., 2018).In this work, we investigate the solvability of this task on general graphs. We give a new, direct proof of the impossibility of approximate agreement on cycles of length c≥4, via a generalisation of Sperner's Lemma to convex polygons. We also extend the reduction from 2-set agreement to a larger class of graphs, showing that approximate agreement on these graphs is unsolvable. On the positive side, we present a wait-free algorithm for a different class of graphs, which properly contains the class of chordal graphs.

Structural Parameterizations with Modulator Oblivion

It is known that problems like Vertex Cover, Feedback Vertex Set and Odd Cycle Transversal are polynomial time solvable in the class of chordal graphs. We consider these problems in a graph that has at most k vertices whose deletion results in a chordal graph when parameterized by k. While this investigation fits naturally into the recent trend of what is called ‘structural parameterizations’, here we assume that the deletion set is not given. One method to solve them is to compute a k-sized or an approximate (f(k) sized, for a function f) chordal vertex deletion set and then use the structural properties of the graph to design an algorithm. This method leads to at least \(k^{{{\mathcal {O}}}(k)}n^{{{\mathcal {O}}}(1)}\) running time when we use the known parameterized or approximation algorithms for finding a k-sized chordal deletion set on an n vertex graph. In this work, we design \(2^{{{\mathcal {O}}}(k)}n^{{{\mathcal {O}}}(1)}\) time algorithms for these problems. Our algorithms do not compute a chordal vertex deletion set (or even an approximate solution). Instead, we construct a tree decomposition of the given graph in \(2^{{{\mathcal {O}}}(k)}n^{{{\mathcal {O}}}(1)}\) time where each bag is a union of four cliques and \({{\mathcal {O}}}(k)\) vertices. We then apply standard dynamic programming algorithms over this special tree decomposition. This special tree decomposition can be of independent interest. Our algorithms are, what are sometimes called permissive in the sense that given an integer k, they detect whether the graph has no chordal vertex deletion set of size at most k or output the special tree decomposition and solve the problem. We also show lower bounds for the problems we deal with under the strong exponential time hypothesis.

Inductive graph invariants and approximation algorithms

We introduce and study an inductively defined analogue [Formula: see text] of any increasing graph invariant [Formula: see text]. An invariant [Formula: see text] is increasing if [Formula: see text] whenever [Formula: see text] is an induced subgraph of [Formula: see text]. This inductive analogue simultaneously generalizes and unifies known notions like degeneracy, inductive independence number, etc., into a single generic notion. For any given increasing [Formula: see text], this gets us several new invariants and many of which are also increasing. It is also shown that [Formula: see text] is the minimum (over all orderings) of a value associated with each ordering. We also explore the possibility of computing [Formula: see text] (and a corresponding optimal vertex ordering) and identify some pairs [Formula: see text] for which [Formula: see text] can be computed efficiently for members of [Formula: see text]. In particular, it includes graphs of bounded [Formula: see text] values. Some specific examples (like the class of chordal graphs) have already been studied extensively. We further extend this new notion by (i) allowing vertex weighted graphs, (ii) allowing [Formula: see text] to take values from a totally ordered universe with a minimum and (iii) allowing the consideration of [Formula: see text]-neighborhoods for arbitrary but fixed [Formula: see text]. Such a generalization is employed in designing efficient approximations of some graph optimization problems. Precisely, we obtain efficient algorithms (by generalizing the known algorithm of Ye and Borodin [Y. Ye and A. Borodin, Elimination graphs, ACM Trans. Algorithms 8(2) (2012) 1–23] for special cases) for approximating optimal weighted induced [Formula: see text]-subgraphs and optimal [Formula: see text]-colorings (for hereditary [Formula: see text]’s) within multiplicative factors of (essentially) [Formula: see text] and [Formula: see text] respectively, where [Formula: see text] denotes the inductive analogue (as defined in this work) of optimal size of an unweighted induced [Formula: see text]-subgraph of the input and [Formula: see text] is the minimum size of a forbidden induced subgraph of [Formula: see text]. Our results generalize the previous result on efficiently approximating maximum independent sets and minimum colorings on graphs of bounded inductive independence number to optimal [Formula: see text]-subgraphs and [Formula: see text]-colorings for arbitrary hereditary classes [Formula: see text]. As a corollary, it is also shown that any maximal [Formula: see text]-subgraph approximates an optimal solution within a factor of [Formula: see text] for unweighted graphs, where [Formula: see text] is maximum size of any induced [Formula: see text]-subgraph in any local neighborhood [Formula: see text].

On the effectiveness of the incremental approach to minimal chordal edge modification

Because edge modification problems are computationally difficult for most target graph classes, considerable attention has been devoted to inclusion-minimal edge modifications, which are usually polynomial-time computable and which can serve as an approximation of minimum cardinality edge modifications, albeit with no guarantee on the cardinality of the resulting modification set. Over the past fifteen years, the primary design approach used for inclusion-minimal edge modification algorithms is based on a specific incremental scheme. Unfortunately, nothing guarantees that the set E of edge modifications of a graph G that can be obtained in this specific way spans all the inclusion-minimal edge modifications of G. Here, we focus on edge modification problems into the class of chordal graphs and we show that for this the set E may not even contain any solution of minimum size and may not even contain a solution close to the minimum; in fact, we show that it may not contain a solution better than within an Ω(n) factor of the minimum. These results show strong limitations on the use of the current favored algorithmic approach to inclusion-minimal edge modification in heuristics for computing a minimum cardinality edge modification. They suggest that further developments might be better using other approaches.

New algorithms for weighted k-domination and total k-domination problems in proper interval graphs

Given a positive integer k, a k-dominating set in a graph G is a set of vertices such that every vertex not in the set has at least k neighbors in the set. A total k-dominating set is a set of vertices such that every vertex of the graph has at least k neighbors in the set. The problems of finding the minimum size of a k-dominating, respectively total k-dominating set, in a given graph, are referred to as k-domination, respectively total k-domination. These generalizations of the classical domination and total domination problems are known to be NP-hard in the class of chordal graphs, and, more specifically, even in the classes of split graphs (both problems) and undirected path graphs (in the case of total k-domination). On the other hand, it follows from previous works by Bui-Xuan et al. (2013) [8] and by Belmonte and Vatshelle (2013) [3] that these two families of problems are solvable in time O(|V(G)|3k+4) in the class of interval graphs. We develop faster algorithms for k-domination and total k-domination in the class of proper interval graphs, by means of reduction to a single shortest path computation in a derived directed acyclic graph with O(|V(G)|2k) nodes and O(|V(G)|4k) arcs. We show that a suitable implementation, which avoids constructing all arcs of the digraph, leads to a running time of O(|V(G)|3k). The algorithms are also applicable to the weighted case.

Detecting strong cliques

A strong clique in a graph is a clique intersecting every maximal independent set. We study the computational complexity of six algorithmic decision problems related to strong cliques in graphs and almost completely determine their complexity in the classes of chordal graphs, weakly chordal graphs, line graphs and their complements, and graphs of maximum degree at most three. Our results rely on connections with matchings and relate to several graph properties studied in the literature, including well-covered graphs, localizable graphs, and general partition graphs.

Linear separation of connected dominating sets in graphs

A connected dominating set in a graph is a dominating set of vertices that induces a connected subgraph. Following analogous studies in the literature related to independent sets, dominating sets, and total dominating sets, we study in this paper the class of graphs in which the connected dominating sets can be separated from the other vertex subsets by a linear weight function. More precisely, we say that a graph is connected-domishold if it admits non-negative real weights associated to its vertices such that a set of vertices is a connected dominating set if and only if the sum of the corresponding weights exceeds a certain threshold. We characterize the graphs in this non-hereditary class in terms of a property of the set of minimal cutsets of the graph. We give several characterizations for the hereditary case, that is, when each connected induced subgraph is required to be connected-domishold. The characterization by forbidden induced subgraphs implies that the class properly generalizes two well known classes of chordal graphs, the block graphs and the trivially perfect graphs. Finally, we study certain algorithmic aspects of connected-domishold graphs. Building on connections with minimal cutsets and properties of the derived hypergraphs and Boolean functions, we show that our approach leads to new polynomially solvable cases of the weighted connected dominating set problem.

A decomposition approach to solve the selective graph coloring problem in some perfect graph families

Graph coloring is the problem of assigning a minimum number of colors to all vertices of a graph such that no two adjacent vertices receive the same color. The selective graph coloring problem is a generalization of the standard graph coloring problem; given a graph with a partition of its vertex set into clusters, the objective is to choose exactly one vertex per cluster so that, among all possible selections, the number of colors necessary to color the vertices in the selection is minimum. This study focuses on a decomposition based exact solution framework for selective coloring in some perfect graph families: in particular, permutation, generalized split, and chordal graphs where the selective coloring problem is known to be NP‐hard. Our method combines integer programming techniques and combinatorial algorithms for the graph classes of interest. We test our method on graphs with different sizes and densities, present computational results and compare them with solving an integer programming formulation of the problem by CPLEX, and a state‐of‐the art algorithm from the literature. Our computational experiments indicate that our decomposition approach significantly improves solution performance in low‐density graphs, and regardless of edge‐density in the class of chordal graphs.

Bounding the Clique‐Width of H‐Free Chordal Graphs

AbstractA graph is H‐free if it has no induced subgraph isomorphic to H. Brandstädt, Engelfriet, Le, and Lozin proved that the class of chordal graphs with independence number at most 3 has unbounded clique‐width. Brandstädt, Le, and Mosca erroneously claimed that the gem and co‐gem are the only two 1‐vertex P4‐extensions H for which the class of H‐free chordal graphs has bounded clique‐width. In fact we prove that bull‐free chordal and co‐chair‐free chordal graphs have clique‐width at most 3 and 4, respectively. In particular, we find four new classes of H‐free chordal graphs of bounded clique‐width. Our main result, obtained by combining new and known results, provides a classification of all but two stubborn cases, that is, with two potential exceptions we determine all graphs H for which the class of H‐free chordal graphs has bounded clique‐width. We illustrate the usefulness of this classification for classifying other types of graph classes by proving that the class of ‐free graphs has bounded clique‐width via a reduction to K4‐free chordal graphs. Finally, we give a complete classification of the (un)boundedness of clique‐width of H‐free weakly chordal graphs.

Join colourings of chordal graphs

We consider certain partition problems typified by the following two questions. When is a given graph the join of two k-colourable graphs? When is a given graph the join of a k-colourable graph and a graph whose complement is k-colourable? We focus on the class of chordal graphs, and employ the language of matrix partitions. Our emphasis is on forbidden induced subgraph characterizations. We describe all chordal minimal obstructions to the existence of such partitions; they are surprisingly small and regular. We also give another characterization which yields a more efficient recognition algorithm. By contrast, most of these partition problems are NP-complete if chordality is not assumed.

On probe 2-clique graphs and probe diamond-free graphs

Graph Theory Given a class G of graphs, probe G graphs are defined as follows. A graph G is probe G if there exists a partition of its vertices into a set of probe vertices and a stable set of nonprobe vertices in such a way that non-edges of G, whose endpoints are nonprobe vertices, can be added so that the resulting graph belongs to G. We investigate probe 2-clique graphs and probe diamond-free graphs. For probe 2-clique graphs, we present a polynomial-time recognition algorithm. Probe diamond-free graphs are characterized by minimal forbidden induced subgraphs. As a by-product, it is proved that the class of probe block graphs is the intersection between the classes of chordal graphs and probe diamond-free graphs.

Formula omitted]-domination for chordal graphs and related graph classes

In this work we obtain a new graph class where the {k}-dominating function problem ({k}-DOM) is NP-complete: the class of chordal graphs. We also identify some maximal subclasses for which it is polynomial time solvable. Firstly, by relating this problem with the k-dominating function problem (k-DOM), we prove that {k}-DOM is polynomial time solvable for strongly chordal graphs. Besides, by expressing the property involved in k-DOM in Counting Monadic Second-order Logic, we obtain that both problems are linear time solvable for bounded tree-width graphs. Finally, we show that {k}-DOM is linear time solvable for spider graphs.

Special Eccentric Vertices for the Class of Chordal Graphs and Related Classes

A vertex is simplicial if the vertices of its neighborhood are pairwise adjacent. It is known that, for every vertex v of a chordal graph, there exists a simplicial vertex among the vertices at maximum distance from v. Here we prove similar properties in other classes of graphs related to that of chordal graphs. Those properties will not be in terms of simplicial vertices, but in terms of other types of vertices that are used to characterize those classes.

Partitioning Chordal Graphs

Matrix partition problems generalize graph colouring and homomorphism problems and occur frequently in the study of perfect graphs. It is difficult to decide, even for a small matrix M, whether the M-partition problem is polynomial time solvable or NP-complete (or possibly neither), and whether M-partitionable graphs can be characterized by a finite set of forbidden induced subgraphs (or perhaps some other first order condition). We discuss these problems for the class of chordal graphs.

Equistable graphs, general partition graphs, triangle graphs, and graph products

In this paper we examine the connections between equistable graphs, general partition graphs and triangle graphs. While every general partition graph is equistable and every equistable graph is a triangle graph, not every triangle graph is equistable, and a conjecture due to Jim Orlin states that every equistable graph is a general partition graph. The conjecture holds within the class of chordal graphs; if true in general, it would provide a combinatorial characterization of equistable graphs. Exploiting the combinatorial features of triangle graphs and general partition graphs, we verify Orlin’s conjecture for several graph classes, including AT-free graphs and various product graphs. More specifically, we obtain a complete characterization of the equistable graphs that are non-prime with respect to the Cartesian or the tensor product, and provide some necessary and sufficient conditions for the equistability of strong, lexicographic and deleted lexicographic products. We also show that the general partition graphs are not closed under the strong product, answering a question by McAvaney et al.

Optimization problems in multiple subtree graphs

We study various optimization problems in t-subtree graphs, the intersection graphs of t -subtrees, where a t -subtree is the union of t disjoint subtrees of some tree. This graph class generalizes both the class of chordal graphs and the class of t -interval graphs, a generalization of interval graphs that has recently been studied from a combinatorial optimization point of view. We present approximation algorithms for the Maximum Independent Set, Minimum Coloring, Minimum Vertex Cover, Minimum Dominating Set, and Maximum Clique problems.

Recognizing edge clique graphs among interval graphs and probe interval graphs

The edge clique graph of a graph H is the one having the edge set of H as vertex set, two vertices being adjacent if and only if the corresponding edges belong to a common complete subgraph of H . We characterize the graph classes {edge clique graphs} ∩ {interval graphs} as well as {edge clique graphs} ∩ {probe interval graphs}, which leads to polynomial time recognition algorithms for them. This work generalizes corresponding results in [M.R. Cerioli, J.L. Szwarcfiter, Edge clique graphs and some classes of chordal graphs, Discrete Math. 242 (2002) 31–39].

Generating Chordal Graphs Included in Given Graphs

A chordal graph is a graph which contains no chordless cycle of at least four edges as an induced subgraph. The class of chordal graphs contains many famous graph classes such as trees, interval graphs, and split graphs, and is also a subclass of perfect graphs. In this paper, we address the problem of enumerating all labeled chordal graphs included in a given graph. We think of some variations of this problem. First we introduce an algorithm to enumerate all connected labeled chordal graphs in a complete graph of n vertices. Next, we extend the algorithm to an algorithm to enumerate all labeled chordal graphs in a n-vertices complete graph. Then, we show that we can use, with small changes, these algorithms to generate all (connected or not necessarily connected) labeled chordal graphs in arbitrary graph. All our algorithms are based on reverse search method, and time complexities to generate a chordal graph are O(1), and also O(1) delay. Additionally, we present an algorithm to generate every clique of a given chordal graph in constant time. Using these algorithms we obtain combinatorial Gray code like sequences for these graph structures in which the differences between two consecutive graphs are bounded by a constant size.