Claw-free Graphs Research Articles

On the Complexity of Local-Equitable Coloring in Claw-Free Graphs with Small Degree

On the Complexity of Local-Equitable Coloring in Claw-Free Graphs with Small Degree

Carathéodory Number of P 3 -Convexity of Claw-Free Graphs

Given a connected simple undirected graph G = ( V , E ) , a subset S of V is P 3 -convex if each vertex of G not in S has at most one neighbor in S . The P 3 -convex hull ⟨ S ⟩ of S is the smallest P 3 -convex set containing S . A Carathéodory set of G is a set S ⊆ V such that ⟨ S ⟩ ∖ ⋃ w ∈ S ⟨ S ∖ { w } ⟩ is non-empty. The Carathéodory number of G , denoted by C ( G ) , is the largest cardinality of a Carathéodory set of G . In this paper, we settle the conjecture posed by Barbosa et al. appeared in [SIAM J. Discrete Math. 26 (2012) 929–939] in the affirmative, which states that for a claw-free graph G of order n ( G ) , the Carathéodory number C ( G ) of the P 3 -convexity satisfies C ( G ) ≤ 2 n ( G ) + 6 5 . Furthermore, we determine all graphs attaining the bound.

On 2-factors with long cycles in 3-connected claw-free graphs

For a graph H, let δ(H) be the minimum degree and let c(H) be the length of a longest cycle in H. A 2-factor of H is a spanning subgraph of H in which every component is a cycle. In [Discrete Math. 313 (2013) 1934-1943], Čada and Chiba asked if H is a 3-connected claw-free graph of order n with δ(H) sufficiently large, does H have a 2-factor F such that c(F)≥min⁡{9δ(H)−3,n}? In this paper, to answer their question, we show that if H is a 3-connected claw-free graph of order n with δ(H)≥15, then H has a 2-factor F such that c(F)≥min⁡{9δ(H)−3,n}.

Counting kernels in directed graphs with arbitrary orientations

A kernel of a directed graph is a subset of vertices that is both independent and absorbing (every vertex not in the kernel has an out-neighbour in the kernel).Not all directed graphs contain kernels, and computing a kernel or deciding that none exist is NP-complete even on low-degree planar digraphs. The existing polynomial-time algorithms for this problem are all obtained by restricting both the undirected structure and the edge orientations of the input: for example, to chordal graphs without bidirectional edges (Pass-Lanneau, Igarashi and Meunier, Discrete Appl Math 2020) or to permutation graphs where each clique has a sink (Abbas and Saoula, 4OR 2005).By contrast, we count the kernels of a fuzzy circular interval graph in polynomial time, regardless of its edge orientations, and return a kernel when one exists. (Fuzzy circular interval graphs were introduced by Chudnovsky and Seymour in their structure theorem for claw-free graphs.)We also consider kernels on cographs, where we establish NP-hardness in general but linear-time solvability on the subclass of threshold graphs.

On claw-free graphs with all but four eigenvalues equal to 0 or [formula omitted]

A graph is claw-free if it does not contain a star of order 4 as an induced subgraph. In this paper, we characterize all claw-free connected graphs with all but four eigenvalues equal to 0 or −1.

Computing Metric Dimension of Two Types of Claw-free Cubic Graphs with Applications

Consider the simple connected graph G with vertex set V(G) and edge set E(G). A graph \(G\) can be resolved by \(R\) if each vertex’s representation of distances to the other vertices in \(R\) uniquely identifies it. The minimum cardinality of the set \(R\) is the metric dimension of \(G\). The length of the shortest path between any two vertices, x, y in V(G), is signified by the distance symbol d(x, y). An ordered k-tuple \(r(x/R)=(d(x,z_1),d(x,\ z_2),…,d(x,z_k))\) represents representation of \(x\) with respect to \(R\) for an ordered subset \(R={\{z}_1,z_2,z_3…,z_k\}\) of vertices and vertex \(x\) in a connected graph. Metric dimension is used in a wide range of contexts where connection, distance, and connectedness are essential factors. It facilitates understanding the structure and dynamics of complex networks and problems relating to robotics network design, navigation, optimization, and facility location. Robots can optimize their localization and navigation methods using a small number of reference sites due to the pertinent idea of metric dimension. As a result, many robotic applications, such as collaborative robotics, autonomous navigation, and environment mapping, are more accurate, efficient, and resilient. A claw-free cubic graph (CCG) is one in which no induced subgraph is a claw. CCG proves helpful in various fields, including optimization, network design, and algorithm development. They offer intriguing structural and algorithmic properties. Developing algorithms and results for claw-free graphs frequently has applications in solving of challenging real-world situations. The metric dimension of a couple of claw-free cubic graphs (CCG), a string of diamonds (SOD), and a ring of diamonds (ROD) will be determined in this work.

The Realizability of Theta Graphs as Reconfiguration Graphs of Minimum Independent Dominating Sets

Abstract The independent domination number i(G) of a graph G is the minimum cardinality of a maximal independent set of G, also called an i(G)-set. The i-graph of G, denoted ℐ (G), is the graph whose vertices correspond to the i(G)-sets, and where two i(G)-sets are adjacent if and only if they differ by two adjacent vertices. Not all graphs are i-graph realizable, that is, given a target graph H, there does not necessarily exist a source graph G such that H = ℐ (G). We consider a class of graphs called “theta graphs”: a theta graph is the union of three internally disjoint nontrivial paths with the same two distinct end vertices. We characterize theta graphs that are i-graph realizable, showing that there are only finitely many that are not. We also characterize those line graphs and claw-free graphs that are i-graphs, and show that all 3-connected cubic bipartite planar graphs are i-graphs.

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.

Computing well-covered vector spaces of graphs using modular decomposition

A graph is well-covered if all its maximal independent sets have the same cardinality. This concept was introduced by Plummer in 1970 and naturally generalizes to the weighted case. Given a graph G, a real-valued vertex weight function w is said to be a well-covered weighting of G if all its maximal independent sets are of the same weight with respect to w. The set of all well-covered weightings of a graph G forms a vector space over the field of real numbers, called the well-covered vector space of G. Since the problem of recognizing well-covered graphs is co\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{co}$$\\end{document}-NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{NP}$$\\end{document}-complete, the problem of computing the well-covered vector space of a given graph is co\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{co}$$\\end{document}-NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{NP}$$\\end{document}-hard. Levit and Tankus showed in 2015 that the problem admits a polynomial-time algorithm in the class of claw-free graphs. In this paper, we give two general reductions for the problem, one based on anti-neighborhoods and one based on modular decomposition, combined with Gaussian elimination. Building on these results, we develop a polynomial-time algorithm for computing the well-covered vector space of a given fork-free graph, generalizing the result of Levit and Tankus. Our approach implies a polynomial-time recognition algorithm for the class of well-covered fork-free graphs and also generalizes some known results on cographs.

Counterexamples to the Characterisation of Graphs with Equal Independence and Annihilation Number

In this paper, we disprove the claimed characterisation of graphs with equal independence and annihilation number as proposed by Larson and Pepper [Electron. J. Comb. 2011]. The annihilation number of a graph is defined as the largest integer $p$ such that the sum of its smallest $p$ degrees is greater than or equal to its size, i.e., its number of edges. Larson and Pepper claimed that for a given graph $G=(V,E)$, its independence number $\alpha(G)$ equals its annihilation number $a(G)$ if and only if $$\begin{array}{ll}(1)~~ a(G)\geq \frac n2:& \alpha'(G)=a(G)\\[2mm](2)~~ a(G)= \frac{n-1}{2}:& \alpha'(G-v)=a(G) ~\text{ for some } v\in V.\end{array}$$This paper provides series of counterexamples with an arbitrarily large number of vertices, an arbitrarily large number of components, an arbitrarily large independence number, and an arbitrarily large difference between the critical and the regular independence number. Furthermore, we identify the error in the proof of Larson and Pepper's theorem. Yet, we show that the theorem still holds for bipartite graphs and connected claw-free graphs.

Domination versus edge domination on claw-free graphs

When G is a (finite and simple) graph, we prove that its domination number is at most its edge-domination number if G is a claw-free graph with minimum degree at least two. That generalizes an earlier result of Baste et al. (2020) on cubic claw-free graphs.

Hamilton-connected claw-free graphs with Ore-degree conditions

Let H be a 3-connected claw-free graph on n vertices, and define σ2(H)=min{dH(u)+dH(v):uv∉E(H)}. Kužel et al. proved that H is Hamilton-connected if n≥142 and δ(H)≥n+508. Let HU be an UM-closure of H. We generalize the result above with degree sum conditions by proving the following.(i) For any positive integer p and real number ϵ, there exist an integer N(p,ϵ)=8p2−(2p+1)|ϵ|−2p>0 and a family Gp(r), which can be generated by a finite number of graphs of order r≤max{8,6p+3}, such that if n>N(p,ϵ) and σ2(H)≥n+ϵp(or δ(H)≥n+ϵ2p), then H is Hamilton-connected if and only if HU=L(G) for G∉Gp(r).(ii) If n≥85 and σ2(H)≥n+44(or δ(H)≥n+48), then either H is Hamilton-connected or HU=L(V8′), where V8′ is the graph obtained from the Wagner graph V8 by attaching |E(V8′)|−128 pendant edges at each vertex of V8.

Factors with Red–Blue Coloring of Claw-Free Graphs and Cubic Graphs

Factors with Red–Blue Coloring of Claw-Free Graphs and Cubic Graphs

Further results on outer independent 2-rainbow dominating functions of graphs

Let G = (V(G), E(G)) be a graph. A function f : V(G) → ℙ({1, 2}) is a 2-rainbow dominating function if for every vertex v with f(v) = ∅, f(N(v)) = {1, 2}. An outer-independent 2-rainbow dominating function (OI2RD function) of G is a 2-rainbow dominating function f for which the set of all v 2208 V(G) with f(v) = ∅ is independent. The outer independent 2-rainbow domination number (OI2RD number) γoir2(G) is the minimum weight of an OI2RD function of G. In this paper, we first prove that n/2 is a lower bound on the OI2RD number of a connected claw-free graph of order n and characterize all such graphs for which the equality holds, solving an open problem given in an earlier paper. In addition, a study of this parameter for some graph products is carried out. In particular, we give a closed (resp. an exact) formula for the OI2RD number of rooted (resp. corona) product graphs and prove upper bounds on this parameter for the Cartesian product and direct product of two graphs.

Locally irregular edge-coloring of subcubic graphs

A graph is locally irregular if no two adjacent vertices have the same degree. A locally irregular edge-coloring of a graph G is such an (improper) edge-coloring that the edges of any fixed color induce a locally irregular subgraph. Among the graphs admitting a locally irregular edge-coloring, i.e., decomposable graphs, only one is known to require 4 colors, while for all the others it is believed that 3 colors suffice. In this paper, we prove that decomposable claw-free graphs with maximum degree 3, all cycle permutation graphs, and all generalized Petersen graphs admit a locally irregular edge-coloring with at most 3 colors. We also discuss when 2 colors suffice for a locally irregular edge-coloring of cubic graphs and present an infinite family of cubic graphs of girth 4 which require 3 colors.

Spanning Trees of a Claw-Free Graph Whose Reducible Stems Have Few Leaves

Let T be a tree. The reducible stem of T is the smallest subtree that contains all branch vertices of T. In this paper, we first use a new technique of Gould and Shull [5] to state a new short proof for a result of Kano et al. [10] on the spanning tree with a bounded number of leaves in a claw-free graph. After that, we use a similar idea to prove a sharp sufficient condition for a claw-free graph having a spanning tree whose reducible stem has few leaves.

Disjunctive domination in graphs with minimum degree at least two

A set D of vertices in G is a disjunctive dominating set in G if every vertex not in D is adjacent to a vertex of D or has at least two vertices in D at distance 2 from it in G. The disjunctive domination number, γ2d(G), of G is the minimum cardinality of a disjunctive dominating set in G. In this paper, we show that if G be a graph of order at least 3, δ(G)≥2 and with no component isomorphic to any of eight forbidden graphs, then γ2d(G)≤|G|3. Moreover, we provide an infinite family of graphs attaining this bound. In addition, we also study the case that G is a claw-free graph with minimum degree at least two.

On the minimum degree of minimally 1-tough, triangle-free graphs and minimally 3/2-tough, claw-free graphs

A graph G is minimally t-tough if the toughness of G is t and deletion of any edge from G decreases its toughness. Katona et al. conjectured that the minimum degree of any minimally t-tough graph is ⌈2t⌉ and gave some upper bounds on the minimum degree of the minimally t-tough graphs in [5,6]. In this paper, we show that a minimally 1-tough graph G with girth g≥5 has minimum degree at most ⌊ng+1⌋+g−1, and a minimally 1-tough graph with girth 4 has minimum degree at most n+64. We also prove that the minimum degree of minimally 32-tough, claw-free graphs is 3.

Absence of zeros implies strong spatial mixing

In this paper we show that absence of complex zeros of the partition function of the hard-core model on any family of bounded degree graphs that is closed under taking induced subgraphs implies that the associated probability measure, the hard-core measure, satisfies strong spatial mixing on that family. As a corollary we obtain that the hard-core measure on the family of bounded degree claw-free graphs satisfies strong spatial mixing for every value of the fugacity parameter. We furthermore derive strong spatial mixing for graph homomorphism measures from absence of zeros of the graph homomorphism partition function.

Spanning trees with few peripheral branch vertices in a connected claw-free graph

Spanning trees with few peripheral branch vertices in a connected claw-free graph