Showing papers on "Chordal graph published in 2022"

Bounding twin-width for bounded-treewidth graphs, planar graphs, and bipartite graphs

Abstract: Twin-width is a newly introduced graph width parameter that aims at generalizing a wide range of “nicely structured” graph classes. In this work, we focus on obtaining good bounds on twin-width $$\textbf{tww}(G)$$ for graphs G from a number of classic graph classes. We prove the following: An important idea behind the bounds for planar graphs is to use an embedding of the graph and sphere-cut decompositions to obtain good bounds on neighbourhood complexity.

On maximal cliques of Cayley graphs over fields

Abstract: We describe a new class of maximal cliques, with a vector space structure, of Cayley graphs defined on the additive group of a field. In particular, we show that in the cubic Paley graph with order $q^3$, the subfield with $q$ elements forms a maximal clique. Similar statements also hold for quadruple Paley graphs and Peisert graphs with quartic order.

A Note of Generalization of Fractional ID-factor-critical Graphs

Abstract: In communication networks, the binding numbers of graphs (or networks) are often used to measure the vulnerability and robustness of graphs (or networks). Furthermore, the fractional factors of graphs and the fractional ID-[a, b]-factor-critical covered graphs have a great deal of important applications in the data transmission networks. In this paper, we investigate the relationship between the binding numbers of graphs and the fractional ID-[a, b]-factor-critical covered graphs, and derive a binding number condition for a graph to be fractional ID-[a, b]-factor-critical covered, which is an extension of Zhou’s previous result [S. Zhou, Binding numbers for fractional ID-k-factor-critical graphs, Acta Mathematica Sinica, English Series 30(1)(2014)181–186].

Polynomial-time Algorithm for Maximum Weight Independent Set on <i>P</i> ₆ -free Graphs

Abstract: In the classic Maximum Weight Independent Set problem, we are given a graph G with a nonnegative weight function on its vertices, and the goal is to find an independent set in G of maximum possible weight. While the problem is NP-hard in general, we give a polynomial-time algorithm working on any P 6 -free graph, that is, a graph that has no path on 6 vertices as an induced subgraph. This improves the polynomial-time algorithm on P 5 -free graphs of Lokshtanov et al. [ 15 ] and the quasipolynomial-time algorithm on P 6 -free graphs of Lokshtanov et al. [ 14 ]. The main technical contribution leading to our main result is enumeration of a polynomial-size family ℱ of vertex subsets with the following property: For every maximal independent set I in the graph, ℱ contains all maximal cliques of some minimal chordal completion of G that does not add any edge incident to a vertex of I .

Regularity of symbolic powers of edge ideals of chordal graphs

Abstract: Assume that G is a chordal graph with edge ideal I(G) and induced matching number ν(G). We denote the sth symbolic power of I(G) by I(G)(s). It is shown that reg(I(G)(s))=2s+ν(G)−1 for every integer s≥1.

Testing Isomorphism of Chordal Graphs of Bounded Leafage is Fixed-Parameter Tractable (Extended Abstract)

Abstract: AbstractThe computational complexity of the graph isomorphism problem is considered to be a major open problem in theoretical computer science. It is known that testing isomorphism of chordal graphs is polynomial-time equivalent to the general graph isomorphism problem. Every chordal graph can be represented as the intersection graph of some subtrees of a representing tree, and the leafage of a chordal graph is defined to be the minimum number of leaves in a representing tree for it. We prove that chordal graph isomorphism is fixed parameter tractable with leafage as parameter.Keywordsgraph isomorphismchordal graphsleafagefixed parameter tractable problem

A divide-and-conquer approach for reconstruction of {C≥5}-free graphs via betweenness queries

Abstract: We study the query complexity of reconstructing {C≥5}-free graphs with respect to the betweenness oracle. In particular, we show that hidden {C≥5}-free graphs can be reconstructed by using O(Δ14⋅log2⁡n+Δ9⋅nlog2⁡n) betweenness queries in expectation, where Δ denotes the maximum degree of the given graph and n denotes the number of vertices. In addition, we propose two improved randomized algorithms for two subclasses of {C≥5}-free graphs, namely the distance-hereditary graphs and the chordal graphs. For the former class, our algorithm uses O(Δ10⋅log2⁡n+Δ5⋅nlog2⁡n) betweenness queries in expectation, and for the latter class, our algorithm uses O(Δ2⋅nlog2⁡n) betweenness queries in expectation.

Studies of Chordal Ring Networks via Double Metric Dimensions

Abstract: Locating the sources of information spreading in networks including tracking down the origins of epidemics, rumors in social networks, and online computer viruses, has a wide range of applications. In many situations, identifying where an epidemic started, or which node in a network served as the source, is crucial. But it can be difficult to determine the root of an outbreak, especially when data are scarce and noisy. The goal is to find the source of the infection by analysing data provided by only a limited number of observers, such as nodes that can indicate when and where they are infected. Our goal is to investigate where the least number of observers should be placed, which is similar to how to figure out the minimal doubly resolving sets in the network. In this paper, we calculate the double metric dimension of chordal ring networks C R n 1,3,5 by describing their minimal doubly resolving sets.

Twin-width and Transductions of Proper k-Mixed-Thin Graphs

Abstract: AbstractThe new graph parameter twin-width, recently introduced by Bonnet et al., allows for an FPT algorithm for testing all FO properties of graphs. This makes classes of efficiently bounded twin-width attractive from the algorithmic point of view. In particular, such classes (of small twin-width) include proper interval graphs, and (as digraphs) posets of width k. Inspired by an existing generalization of interval graphs into so-called k-thin graphs, we define a new class of proper k-mixed-thin graphs which largely generalizes proper interval graphs. We prove that proper k-mixed-thin graphs have twin-width linear in k, and that a certain subclass of k-mixed-thin graphs is transduction-equivalent to posets of width \(k'\) such that there is a quadratic relation between k and \(k'\).Keywordstwin-widthproper interval graphproper mixed-thin graphtransduction equivalence

On the Maximum Number of Edges in Chordal Graphs of Bounded Degree and Matching Number

Abstract: We determine the maximum number of edges that a chordal graph G can have if its degree, $$\varDelta (G)$$ , and its matching number, $$ u (G)$$ , are bounded. To do so, we show that for every $$d, u \in \mathbb {N}$$ , there exists a chordal graph G with $$\varDelta (G)

Aligning random graphs with a sub-tree similarity message-passing algorithm

Abstract: Abstract The problem of aligning Erdős–Rényi random graphs is a noisy, average-case version of the graph isomorphism problem, in which a pair of correlated random graphs is observed through a random permutation of their vertices. We study a polynomial time message-passing algorithm devised to solve the inference problem of partially recovering the hidden permutation, in the sparse regime with constant average degrees. We perform extensive numerical simulations to determine the range of parameters in which this algorithm achieves partial recovery. We also introduce a generalized ensemble of correlated random graphs with prescribed degree distributions, and extend the algorithm to this case.

Isomorphism Testing for T-graphs in FPT

Abstract: A T-graph (a special case of a chordal graph) is the intersection graph of connected subtrees of a suitable subdivision of a fixed tree T. We deal with the isomorphism problem for T-graphs which is GI-complete in general – when T is a part of the input and even a star. We prove that the T-graph isomorphism problem is in FPT when T is the fixed parameter of the problem. This can equivalently be stated that isomorphism is in FPT for chordal graphs of (so-called) bounded leafage. While the recognition problem for T-graphs is not known to be in FPT wrt. T, we do not need a T-representation to be given (a promise is enough). To obtain the result, we combine a suitable isomorphism-invariant decomposition of T-graphs with the classical tower-of-groups algorithm of Babai, and reuse some of the ideas of our isomorphism algorithm for $$S_d$$ -graphs [MFCS 2020].

An efficient algorithm for counting Markov equivalent DAGs

Abstract: We consider the problem of counting the number of DAGs which are Markov equivalent, i.e., which encode the same conditional independencies between random variables. The problem has been studied, among others, in the context of causal discovery, and it is known that it reduces to counting the number of so-called moral acyclic orientations of certain undirected graphs, notably chordal graphs. Our main empirical contribution is a new algorithm which outperforms previously known exact algorithms for the considered problem by a significant margin. On the theoretical side, we show that our algorithm is guaranteed to run in polynomial time on a broad cubic-time recognisable class of chordal graphs, including interval graphs.

The perfect matching cut problem revisited

Abstract: In a graph, a perfect matching cut is an edge cut that is a perfect matching. perfect matching cut (pmc) is the problem of deciding whether a given graph has a perfect matching cut, and is known to be NP-complete. We revisit the problem and show that pmc remains NP-complete when restricted to bipartite graphs of maximum degree 3 and arbitrarily large girth. Complementing this hardness result, we give two graph classes in which pmc is polynomial-time solvable. The first one includes claw-free graphs and graphs without an induced path on five vertices, the second one properly contains all chordal graphs. Assuming the Exponential Time Hypothesis, we show there is no O⁎(2o(n))-time algorithm for pmc even when restricted to n-vertex bipartite graphs, and also show that pmc can be solved in O⁎(1.2721n) time by means of an exact branching algorithm.

Total 2-Rainbow Domination in Graphs

Abstract: A total k-rainbow dominating function on a graph G=(V,E) is a function f:V(G)→2{1,2,…,k} such that (i) ∪u∈N(v)f(u)={1,2,…,k} for every vertex v with f(v)=∅, (ii) ∪u∈N(v)f(u)≠∅ for f(v)≠∅. The weight of a total 2-rainbow dominating function is denoted by ω(f)=∑v∈V(G)|f(v)|. The total k-rainbow domination number of G is the minimum weight of a total k-rainbow dominating function of G. The minimum total 2-rainbow domination problem (MT2RDP) is to find the total 2-rainbow domination number of the input graph. In this paper, we study the total 2-rainbow domination number of graphs. We prove that the MT2RDP is NP-complete for planar bipartite graphs, chordal bipartite graphs, undirected path graphs and split graphs. Then, a linear-time algorithm is proposed for computing the total k-rainbow domination number of trees. Finally, we study the difference in complexity between MT2RDP and the minimum 2-rainbow domination problem.

Linearizing Partial Search Orders

Abstract: In recent years, questions about the construction of special orderings of a given graph search were studied by several authors. On the one hand, the so called end-vertex problem introduced by Corneil et al. in 2010 asks for search orderings ending in a special vertex. On the other hand, the problem of finding orderings that induce a given search tree was introduced already in the 1980s by Hagerup and received new attention most recently by Beisegel et al. Here, we introduce a generalization of some of these problems by studying the question whether there is a search ordering that is a linear extension of a given partial order on a graph’s vertex set. We show that this problem can be solved in polynomial time on chordal bipartite graphs for LBFS, which also implies the first polynomial-time algorithms for the end-vertex problem and two search tree problems for this combination of graph class and search. Furthermore, we present polynomial-time algorithms for LBFS and MCS on split graphs which generalize known results for the end-vertex and search tree problems.

Weighted Connected Matchings

Abstract: A matching M is a $$\mathscr {P}$$ -matching if the subgraph induced by the endpoints of the edges of M satisfies property $$\mathscr {P}$$ . As examples, for appropriate choices of $$\mathscr {P}$$ , the problems Induced Matching, Uniquely Restricted Matching, Connected Matching and Disconnected Matching arise. For many of these problems, finding a maximum $$\mathscr {P}$$ -matching is a knowingly $$\textsf{NP}$$ -hard problem, with few exceptions, such as Connected Matching, which has the same time complexity as the usual Maximum Matching problem. The weighted variant of Maximum Matching has been studied for decades, with many applications, including the well-known Assignment problem. Motivated by this fact, in addition to some recent research in weighted versions of acyclic and induced matchings, we study the Maximum Weight Connected Matching. In this problem, we want to find a matching M such that the endpoints of its edges induce a connected subgraph and the sum of the edge weights of M is maximum. Unlike the unweighted Connected Matching problem, which is in $$\textsf{P}$$ for general graphs, we show that Maximum Weight Connected Matching is $$\textsf{NP}$$ -hard even for bounded diameter bipartite graphs, starlike graphs, planar bipartite graphs, and subcubic planar graphs, while solvable in linear time for trees and graphs having degree at most two. When we restrict edge weights to be non-negative only, we show that the problem turns out to be polynomially solvable for chordal graphs, while it remains $$\textsf{NP}$$ -hard for most of the other cases. In addition, we consider parameterized complexity. On the positive side, we present a single exponential time algorithm when parameterized by treewidth. As for kernelization, we show that, even when restricted to binary weights, Weighted Connected Matching does not admit a polynomial kernel when parameterized by vertex cover number under standard complexity-theoretical hypotheses.

Heroes in orientations of chordal graphs

Abstract: We characterize all digraphs H such that orientations of chordal graphs with no induced copy of H have bounded dichromatic number.

Feedback Vertex Set and Even Cycle Transversal for $H$-Free Graphs: Finding Large Block Graphs

Abstract: We prove new complexity results for Feedback Vertex Set and Even Cycle Transversal on $H$-free graphs, that is, graphs that do not contain some fixed graph $H$ as an induced subgraph. In particular, we prove that for every $s\geq 1$, both problems are polynomial-time solvable for $sP_3$-free graphs and $(sP_1+P_5)$-free graphs; here, the graph $sP_3$ denotes the disjoint union of $s$ paths on three vertices and the graph $sP_1+P_5$ denotes the disjoint union of $s$ isolated vertices and a path on five vertices. Our new results for Feedback Vertex Set extend all known polynomial-time results for Feedback Vertex Set on $H$-free graphs, namely for $sP_2$-free graphs [Chiarelli et al., Theoret. Comput. Sci., 705 (2018), pp. 75--83], $(sP_1+P_3)$-free graphs [Dabrowski et al., Algorithmica, 82 (2020), pp. 2841--2866] and $P_5$-free graphs [Abrishami et al., Induced subgraphs of bounded treewidth and the container method, in Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, Philadelphia, 2021, pp. 1948--1964]. Together, the new results also show that both problems exhibit the same behavior on $H$-free graphs (subject to some open cases). This is in part due to a new general algorithm we design for finding in a ($sP_3)$-free or $(sP_1+P_5)$-free graph $G$ a largest induced subgraph whose blocks belong to some finite class ${\cal C}$ of graphs. We also compare our results with the state-of-the-art results for the Odd Cycle Transversal problem, which is known to behave differently on $H$-free graphs.