Class Of Line Graphs Research Articles

Formula omitted]-conjectures on the domination game and claw-free graphs

Let γg(G) be the game domination number of a graph G. Rall conjectured that if G is a traceable graph, then γg(G)≤12n(G). Our main result verifies the conjecture over the class of line graphs. Moreover, in this paper we put forward the conjecture that if δ(G)≥2, then γg(G)≤12n(G). We show that both conjectures hold true for claw-free cubic graphs. We further prove the upper bound γg(G)≤1120n(G) over the class of claw-free graphs of minimum degree at least 2. Computer experiments supporting the new conjecture and sharpness examples are also presented.

Conflict‐free chromatic number versus conflict‐free chromatic index

AbstractA vertex coloring of a given graph is conflict‐free if the closed neighborhood of every vertex contains a unique color (i.e., a color appearing only once in the neighborhood). The minimum number of colors in such a coloring is the conflict‐free chromatic number of , denoted . What is the maximum possible conflict‐free chromatic number of a graph with a given maximum degree ? Trivially, , but it is far from optimal—due to results of Glebov, Szabó, and Tardos, and of Bhyravarapu, Kalyanasundaram, and Mathew, the answer is known to be . We show that the answer to the same question in the class of line graphs is —it follows that the extremal value of the conflict‐free chromatic index among graphs with maximum degree is much smaller than the one for conflict‐free chromatic number. The same result for is also provided in the class of near regular graphs, that is, graphs with minimum degree .

From matchings to independent sets

In 1965, Jack Edmonds proposed his celebrated polynomial-time algorithm to find a maximum matching in a graph. It is well-known that finding a maximum matching in G is equivalent to finding a maximum independent set in the line graph of G. For general graphs, the maximum independent set problem is NP-hard. What makes this problem easy in the class of line graphs and what other restrictions can lead to an efficient solution of the problem? In the present paper, we analyse these and related questions. We also review various techniques that may lead to efficient algorithms for the maximum independent set problem in restricted graph families, with a focus given to augmenting graphs and graph transformations. Both techniques have been used in the solution of Edmonds to the maximum matching problem, i.e. in the solution to the maximum independent set problem in the class of line graphs. We survey various results that exploit these techniques beyond the line graphs.

Identifying Codes in Line Graphs

AbstractAn identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbors within the code. We study the edge‐identifying code problem, i.e. the identifying code problem in line graphs. If denotes the size of a minimum identifying code of an identifiable graph G, we show that the usual bound , where n denotes the order of G, can be improved to in the class of line graphs. Moreover, this bound is tight. We also prove that the upper bound , where is the line graph of G, holds (with two exceptions). This implies that a conjecture of R. Klasing, A. Kosowski, A. Raspaud, and the first author holds for a subclass of line graphs. Finally, we show that the edge‐identifying code problem is NP‐complete, even for the class of planar bipartite graphs of maximum degree 3 and arbitrarily large girth.

Edge identifying codes

Abstract We study the edge identifying code problem, i.e. the identifying code problem in line graphs. If γ ID ( G ) denotes the size of a minimum identifying code of a graph G , we show that the usual bound γ ID ( G ) ⩾ ⌈ log 2 ( n + 1 ) ⌉ , where n denotes the order of G , can be improved to Θ ( n ) in the class of line graphs. Moreover this bound is tight. We also prove that the upper bound γ ID ( L ( G ) ) ⩽ 2 ⋅ | V ( G ) | − 4 holds. This implies that a conjecture of R. Klasing, A. Kosowski, A. Raspaud and the first author holds for a subclass of line graphs. Finally, we show that the edge identifying code problem is NP-complete, even for the class of planar bipartite graphs of maximum degree 3 and arbitrarily large girth.

A polynomial algorithm to find an independent set of maximum weight in a fork-free graph

The class of fork-free graphs is an extension of claw-free graphs and their subclass of line graphs. The first polynomial-time solution to the maximum weight independent set problem in the class of line graphs, which is equivalent to the maximum matching problem in general graphs, has been proposed by Edmonds in 1965 and then extended to the entire class of claw-free graphs by Minty in 1980. Recently, Alekseev proposed a solution for the larger class of fork-free graphs, but only for the unweighted version of the problem, i.e., finding an independent set of maximum cardinality. In the present paper, we describe the first polynomial-time algorithm to solve the problem for weighted fork-free graphs.

On Tutte polynomial uniqueness of twisted wheels

A graph G is called T - unique if any other graph having the same Tutte polynomial as G is isomorphic to G . Recently, there has been much interest in determining T -unique graphs and matroids. For example, de Mier and Noy [A. de Mier, M. Noy, On graphs determined by their Tutte polynomials, Graphs Combin. 20 (2004) 105–119; A. de Mier, M. Noy, Tutte uniqueness of line graphs, Discrete Math. 301 (2005) 57–65] showed that wheels, ladders, Möbius ladders, square of cycles, hypercubes, and certain class of line graphs are all T -unique. In this paper, we prove that the twisted wheels are also T -unique.

Augmenting chains in graphs without a skew star

The augmenting chain technique has been applied to solve the maximum stable set problem in the class of line graphs (which coincides with the maximum matching problem) and then has been extended to the class of claw-free graphs. In the present paper, we propose a further generalization of this approach. Specifically, we show how to find an augmenting chain in graphs containing no skew star, i.e. a tree with exactly three vertices of degree 1 of distances 1, 2, 3 from the only vertex of degree 3. As a corollary, we prove that the maximum stable set problem is polynomially solvable in a class that strictly contains claw-free graphs, improving several existing results.

Finding augmenting chains in extensions of claw-free graphs

Finding augmenting chains is in the heart of the maximum matching problem, which is equivalent to the maximum stable set problem in the class of line graphs. Due to the celebrated result of Edmonds, augmenting chains can be found in line graphs in polynomial time. Minty and Sbihi generalized this result to claw-free graphs. In this paper we extend it to larger classes. As a particular consequence, a new polynomially solvable case for the maximum stable set problem has been detected.

Intersection properties of line graphs

Each graph is an intersection graph (intersection multigraph) of a family of sets. Such a family is called a representation if all sets are different and a pseudorepresentation if some are the same. By intersection number we mean the cardinality of the smallest set on which we can construct a representation. So each graph has four intersection numbers. A graph is uniquely representable if it has only one representation on the minimal set. We take into account only triangle free graphs which are uniquely intersectable in any sense and line graphs. We show that line graphs with pendant K 4 - e and clepsydras are line graphs which are not uniquely multiple pseudointersectable (u.m.p.i.) in the class of line graphs.

Construction of a certain class of line graphs

The construction of a line graph is described on the basis of a finite bipartite graph, for a given k > 1. An algorithm is set forth for the construction of the line graph and an example of its use is given.

A polynomial characterization of some graph partitioning problems

The Uniform Partition (UP) and the Simple Max Partition (SMP) problems are NP-complete graph partitioning problems, and polynomial-time algorithms are known for few classes of graphs only. In the present paper, the class of line-graphs is considered and a polynomial algorithm is proposed to solve both problems in this class. When the instance space is extended to digraphs, a characterization is possible which leads to similar results.