Kőnig-Egerváry graphs (KEGs) are the graphs whose maximum size of a matching is equal to the minimum size of a vertex cover. We give an excluded subgraph characterization of KEGs. We show that KEGs are a special case of a more general class of graph: Red/Blue-split graphs, and give an excluded subgraph characterization of Red/Blue-split graphs. We show several consequences of this result includ...

Ž . We present efficient parallel algorithms for two hierarchical clustering heuristics. We point out that these heuristics can also be applied to solving some algorithmic problems in graphs, including split decomposition. We show that efficient parallel split decomposition induces an efficient parallel parity graph recognition algorithm. This is a consequence of the result of S. Cicerone and D...

The notion of a split coloring of a complete graph was introduced by Erdős and Gyárfás [7] as a generalization of split graphs. In this paper, we offer an alternate interpretation by comparing such a coloring to the classical Ramsey coloring problem via a two-round game played against an adversary. We show that the techniques used and bounds obtained on the extremal (r,m)-split coloring problem...

In this presentation, we consider the Split Delivery Vehicle Routing Problem with Time Windows (SDVRPTW) in which the delivery to a customer can be split between any number of vehicles. This problem has not been extensively studied in the literature. The main contribution of this paper is a Branch and Price approach for solving the SDVRPTW without imposing restrictions on the split delivery opt...

We consider two problems, namely Min Split-coloring and Min Cocoloring, that generalize the classical Min Coloring problem by using not only stable sets but also cliques to cover all the vertices of a given graph. We prove the NP-hardness of some cases. We derive approximation results for Min Split-coloring and Min Cocoloring in line graphs, comparability graphs and general graphs. This provide...

Spider graphs are the intersection graphs of subtrees of subdivisions of stars. Thus, spider graphs are chordal graphs that form a common superclass of interval and split graphs. Motivated by previous results on the existence of Hamilton cycles in interval, split and chordal graphs, we show that every 3/2-tough spider graph is hamiltonian. The obtained bound is best possible since there are (3/...

Given a tree and a set P of non-trivial simple paths on it, Vpt(P) is the VPT graph (i.e. the vertex intersection graph) of the paths P of the tree T , and Ept(P) is the EPT graph (i.e. the edge intersection graph) of P. These graphs have been extensively studied in the literature. Given two (edge) intersecting paths in a graph, their split vertices is the set of vertices having degree at least...

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 revisit in this paper the stochastic model for minimum graph-coloring introduced in (C. Murat and V. Th. Paschos, On the probabilistic minimum coloring and minimum k-coloring, Discrete Applied Mathematics 154, 2006), and study the underlying combinatorial optimization problem (called probabilistic coloring) in bipartite and split graphs. We show that the obvious 2-coloring of any connected b...

We investigate the enumerative aspects of various classes of perfect graphs like cographs, split graphs, trivially perfect graphs and threshold graphs. For subclasses of permutation graphs like cographs and threshold graphs we also determine the number of permutations of f1; 2; : : : ; ng such that the permutation graph G] belongs to that class. We establish an interesting bijec-tion between pe...