Complements Of Line Graphs Research Articles

线图的补图的Wiener指标

线图的补图的Wiener指标

Clique-perfectness of complements of line graphs

Abstract The clique-transversal number τc(G) of a graph G is the minimum size of a set of vertices meeting all the cliques. The clique-independence number αc(G) of G is the maximum size of a collection of vertex-disjoint cliques. A graph is clique-perfect if these two numbers are equal for every induced subgraph of G. Unlike perfect graphs, the class of clique-perfect graphs is not closed under graph complementation nor is a characterization by forbidden induced subgraphs known. Nevertheless, partial results in this direction have been obtained. For instance, in [Bonomo, F., M. Chudnovsky and G. Duran, Partial characterizations of clique-perfect graphs I: Subclasses of claw-free graphs, Discrete Appl. Math. 156 (2008), pp. 1058–1082], a characterization of those line graphs that are clique-perfect is given in terms of minimal forbidden induced subgraphs. Our main result is a characterization of those complements of line graphs that are clique-perfect, also by means of minimal forbidden induced subgraphs. This implies an O(n2) time algorithm for deciding the clique-perfectness of complements of line graphs and, for those that are clique-perfect, finding αc and τc.

Super-connected and hyper-connected jump graphs

Let G be a connected graph. The graph G is said to be super-connected if for every minimum vertex cut S of G, G−S has isolated vertices. Moreover, it is said to be hyper-connected if for every minimum vertex cut S, G−S has exactly two components, one of which is an isolated vertex. In this note, we give a necessary and sufficient condition for a graph G whose jump graph J(G) (the complement of line graph of G) is, respectively, super-connected and hyper-connected.

Hamiltonian jump graphs

Let G be a nonempty graph. The jump graph J ( G ) of G is the graph whose vertices are edges of G, and where two vertices of J ( G ) are adjacent if and only if they are not adjacent in G. Equivalently, the jump graph J ( G ) of G is the complement of line graph L ( G ) of G. In this paper, we characterize hamiltonian jump graphs and settle two conjectures posed by Chartrand et al. on jump graphs.

Decomposition of odd-hole-free graphs by double star cutsets and 2-joins

In this paper we decompose odd-hole-free graphs (graphs that do not contain as an induced subgraph a chordless cycle of odd length greater than three) with double star cutsets and 2-joins into bipartite graphs, line graphs of bipartite graphs and the complements of line graphs of bipartite graphs.

Clique-transversal sets of line graphs and complements of line graphs

A clique-transversal set T of a graph G is a set of vertices of G such that T meets all maximal cliques of G. The clique-transversal number, denoted τ c ( G), is the minimum cardinality of a clique-transversal set. Let n be the number of vertices of G. We study classes of graphs G for which n 2 is an upper bound for τ c ( G). Assuming that G has no isolated vertices it is shown that (i) τ c(G)⩽ n 2 for all connected line graphs with the exception of odd cycles, and (ii) τ c(G)⩽ n 2 for all complments of line graphs with the exeption of five small graphs. In addition, a closely related question is studied: call G weakly 2- colorable if its vertices can be colored with 2 colors such that G has no monochromatic maximal clique of size ⩾2. It is proved that a connected line graph G = L( H) is weakly 2-colorable iff H has a 2-coloring of its edges without monochromatic triangles and H is not an odd cycle. Moreover it is shown that complements of line graphs are weakly 2-colorable, with the exception of nine small graphs.