A graph G=(V,E) is word-representable if and only there exists a word w over the alphabet V such that letters x y, x≠y, alternate in xy∈E. split which vertices can be partitioned into clique an independent set. There long line of research on graphs literature, recently, word-representability has attracted interest. In this paper, we first give characterization terms permutations columns adjacen...

The edge tenacity Te(G) of a graph G is dened as:Te(G) = min {[|X|+τ(G-X)]/[ω(G-X)-1]|X ⊆ E(G) and ω(G-X) > 1} where the minimum is taken over every edge-cutset X that separates G into ω(G - X) components, and by τ(G - X) we denote the order of a largest component of G. The objective of this paper is to determine this quantity for split graphs. Let G = (Z; I; E) be a noncomplete connected split...

For a fixed number of colors, we show that, in node-weighted split graphs, cographs, and graphs of bounded tree-width, one can determine in polynomial time whether a proper list-coloring of the vertices of a graph such that the total weight of vertices of each color equals a given value in each part of a fixed partition of the vertices exists. We also show that this result is tight in some sens...

Let be a non-trivial simple graph. A dominating set in graph is of vertices such that every vertex not the adjacent to at least one set. subset minimum neighborhood if and for holds. The cardinality called as number it denoted by . where intersection neighborhoods all small possible, (i.e., ). number, , In other words, smallest needed form neighborhood-dominating concept related study structure...

The Colin de Verdière parameters, μ and ν, are defined to be the maximum nullity of certain real symmetric matrices associated with a given graph. In this work, both of these parameters are calculated for all chordal graphs. For ν the calculation is based solely on maximal cliques, while for μ the calculation depends on split subgraphs. For the case of μ our work extends some recent work on com...

A graph G = (V,E) is called a split graph if there exists a partition V = I ∪K such that the subgraphs G[I ] and G[K] of G induced by I and K are empty and complete graphs, respectively. In this paper, we survey results on the hamiltonian and classification problems for split graphs G with the minimum degree δ(G) ≥ |I | − 4. 2000 Mathematics Subject Classification: 05C45, 05C75.

The Colin de Verdière parameters, μ and ν, are defined to be the maximum nullity of certain real symmetric matrices associated with a given graph. In this work, both of these parameters are calculated for all chordal graphs. For ν the calculation is based solely on maximal cliques, while for μ the calculation depends on split subgraphs. For the case of μ our work extends some recent work on com...