An equivalence graph is a vertex disjoint union of complete graphs. For a graph G, let eq(G) be the irdnimum number of equivalence subgraphs of G needed to cover all edges of G. Similarly, let cc(G) be the minimum number of complete subgraphs of G needed to cover all its edges. Let H be a graph on n vertices with ma,'dmal degree _~d (and minimal degree --~ 1), and let G=I~ be its complement. We...

An f (n) dominance bound on a heuristic for some problem is a guarantee that the heuristic always returns a solution not worse than at least f (n) solutions. In this paper, we analyze several heuristics for Vertex Cover, Set Cover, and Knapsack for dominance bounds. In particular, we show that the well-known maximal matching heuristic of Vertex Cover provides an excellent dominance bound. We in...

Let G be a simple graph with vertex set and edge set   V G   E G . An edge coloring C of G is called an edge cover coloring, if each color appears at least once at each vertex   v V G  . The maximum positive integer k such that G has a k edge cover coloring is called the edge cover chromatic number of G and is denoted by . It is known that for any graph G,   c G    1 c G      ...

In this paper a schema of graphical parameters is proposed. Based on the parameter integrity introduced by Barefoot, Entringer and Swart, members Ψ(G) of this schema have the general form Ψ(G) = min{ |S| + ψ(G − S) : S ⊂ V (G) }, where ψ(G) is another given graphical parameter. Examples include integrity, mean integrity, connectivity and vertex cover number. General results and bounds for the s...

We show that there exists a Martin-Löf random degree which has a strong minimal cover.

Given two graphs G and H, assume that C = {C1, C2, . . . , Cq} is a clique cover of G and U is a subset of V (H). We introduce a new graph operation called the clique cover product, denoted by G ⋆ HU , as follows: for each clique Ci ∈ C , add a copy of the graph H and join every vertex of Ci to every vertex of U . We prove that the independence polynomial of G ⋆ HU I(G ⋆ H ;x) = I(H;x)I(G; xI(H...

Four fundamental graph problems, Minimum vertex cover, Maximum independent set, Minimum dominating set and Maximum cut, are shown to be APX-complete even for cubic graphs. This means that unless P=NP these problems do not admit any polynomial time approximation scheme on input graphs of degree bounded by three.

In cooperative games, a key question is to find a division of payoffs to coalition members in a fair manner. Nucleolus is one of such solution concepts that provides a stable solution for the grand coalition. We study the computation of the nucleolus of a number of cooperative games, including fractional matching games and fractional edge cover games on general weighted graphs, as well as verte...

Motivated by the need for succinct representations of all “small” transversals (or hitting sets) of a hypergraph of fixed rank, we study the complexity of computing such a representation. Next, the existence of a minimal hitting set of at least a given size arises as a subproblem. We give one algorithm for hypergraphs of any fixed rank, and we largely improve an earlier algorithm (by H. Fernau,...