Our concern is deriving genus distributions of graphs obtained by surgical operations on graphs whose genus distribution is known. One operation in focus here is adding an edge. The other is splitting a vertex, for which the inverse operation is edge-contraction. Our main result is this Splitting Theorem: Let G be a graph and w a 4-valent vertex of G. Let H1, H2, and H3 be the three graphs into...

A graph G has a representation modulo n if there exists an injective map f : V (G) → {0, 1, . . . , n} such that vertices u and v are adjacent if and only if |f(u) − f(v)| is relatively prime to n. The representation number rep(G) is the smallest n such that G has a representation modulo n. We present new results involving representation numbers for stars, split graphs, complements of split gra...

We show how to use split decomposition to compute the weighted clique number and the chromatic number of a graph and we apply these results to some classes of graphs. In particular we present an O(nm) algorithm to compute the chromatic number for all those graphs having a split decomposition in which every prime graph is an induced subgraph of either a Ck or a Ck for some k ≥ 3.

In this article we research the structure of d-convex simple graphs in order to extend the already known classes of graphs of this type. We do this using some new operations and new graphs. We introduce the notion of M-prime graphs and split all d-convex simple graphs into M-prime graphs using the M operation. After that we describe all M-prime graphs we know. Mathematics subject classification...

Distance-hereditary graphs form an important class of graphs, from the theoretical point of view, due to the fact that they are the totally decomposable graphs for the split-decomposition. The previous best enumerative result for these graphs is from Nakano et al. (J. Comp. Sci. Tech., 2007), who have proven that the number of distancehereditary graphs on n vertices is bounded by 2d3.59ne. In t...

This paper deals with the graph isomorphism (GI) problem for two graph classes: chordal bipartite graphs and strongly chordal graphs. It is known that GI problem is GI complete even for some special graph classes including regular graphs, bipartite graphs, chordal graphs, comparability graphs, split graphs, and k-trees with unbounded k. On the other side, the relative complexity of the GI probl...

The simple max-cut problem is as follows: given a graph, find a partition of its vertex set into two disjoint sets, such that the number of edges having one endpoint in each set is as large as possible. A split graph is a graph whose vertex set admits a partition into a stable set and a clique. The simple max-cut decision problem is known to be NP-complete for split graphs. An indifference grap...

In 1966, Gallai asked whether all longest paths in a connected graph have nonempty intersection. The answer to this question is not true general and various counterexamples been found. However, there positive solution Gallai’s for many well-known classes of graphs such as split graphs, series–parallel 2K2-free graphs. Split were proven be Hamiltonian under given toughness conditions. This obser...

Gustedt, J., On the pathwidth of chordal graphs, Discrete Applied Mathematics 45 (1993) 233-248. In this paper we first show that the pathwidth problem for chordal graphs is NP-hard. Then we give polynomial algorithms for subclasses. One of those classes are the k-starlike graphs a generalization of split graphs. The other class are the primitive starlike graphs a class of graphs where the inte...