In many combinatorial situations there is a notion of independence of a set of points. Maximal independent sets can be easily constructed by a greedy algorithm, and it is of interest to determine, for example, if they all have the same size or the same parity. Both of these questions may be formulated by weighting the points with elements of an abelian group, and asking whether all maximal inde...

A graph G is well-covered if every maximal independent set has the same cardinality. Let sk denote the number of independent sets of cardinality k, and define the independence polynomial of G to be S(G, z) = ∑ skz k. This paper develops a new graph theoretic operation called power magnification that preserves well-coveredness and has the effect of multiplying an independence polynomial by zc wh...

A graph G(V, E) is a threshold graph if there exist non-negative reals wv, v ∈ V and t such that for every U ⊆ V , P v∈U wv ≤ t if and only if U is a stable set. The threshold dimension of a graph G(V,E), denoted as t(G), is the smallest integer k such that E can be covered by k threshold spanning subgraphs of G. A permutation graph is a graph that can be represented as the intersection graph o...

Let G be a graph. A set S of vertices in G dominates the graph if every vertex of G is either in S or a neighbor of a vertex in S. Finding a minimal cardinality set which dominates the graph is an NP-complete problem. The graph G is well-dominated if all its minimal dominating sets are of the same cardinality. The complexity status of recognizing well-dominated graphs is not known. We show that...

This paper examines a parameterized problem that we refer to as n− k Graph Coloring, i.e., the problem of determining whether a graph G with n vertices can be colored using n−k colors. As the main result of this paper, we show that there exists a O(kn+k+2) = O(n) algorithm for n− k Graph Coloring for each fixed k. The core technique behind this new parameterized algorithm is kernalization via m...

A fractional [a, b]-factor of a graph G is function h from E(G) to [0, 1] satisfying \(a \le d_G^h(v) b\) for every vertex v G, where \(d_G^h(v) = \sum\limits_{e \in E(v)} {h(e)} \) and E(v) {e uv : u ? V (G)}. called b]-covered if contains with h(e) 1 any edge e G. (a, b, k)-critical covered — Q ? V(G) ?Q? k. In this article, we demonstrate neighborhood condition be covered. Furthermore, claim...

We give a simple proof for Kaneko’s theorem which gives a su2cient and necessary condition for the existence of vertex disjoint paths in a graph, each of length at least two, that altogether cover all vertices of the original graph. Moreover we generalize this theorem and give a formula for the maximum number of vertices that can be covered by such a path system. c © 2004 Elsevier B.V. All righ...

We call a graph matching-covered if every line belongs to a perfect matching. We study the technique of "ear-decompositions" of such graphs. We prove that a non-bipartite matchingcovered graph contains K~ or K2@Ka (the triangular prism). Using this result, we give new characterizations of those graphs whose matching and covering numbers are equal. We apply these results to the theory of r-criti...

The tree partition number of an r-edge-colored graph G, denoted by tr(G), is the minimum number k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex-disjoint monochromatic trees. We determine t2(K(n1, n2, . . . , nk)) of the complete k-partite graph K(n1, n2, . . . , nk). In particular, we prove that t2(K(n,m)) = (m − 2)/2n + 2, whe...