A k-edge-weighting of a graph G is a mapping w : E(G) → {1, 2, . . ., k}. An edge-weighting w induces a vertex coloringfor any edge uv. The current paper studies the parameter µ(G), which is the minimum k for which G has a vertex-coloring k-edgeweighting. Exact values of µ(G) are determined for several classes of graphs, including trees and r-regular bipartite graph with r ≥ 3.

Let G = (V, E) be a graph and let r ≥ 1 be an integer.notes the number of edges in any shortest path between x and y. D is known as an r-identifying code (r-locating-dominating set, respectively), if for all verticesx ∈ V (x ∈ V \D, respectively), D r (x) are all nonempty and different. In this paper, we provide complete results for r-identifying codes in paths and odd cycles; we also give complete results for 2-locating-dominating sets in cycles.

Let G = (V, E) be a graph without isolated vertices. A set S ⊆ V is a paired-domination set if every vertex in V − S is adjacent to a vertex in S and the subgraph induced by S contains a perfect matching. The paired-domination problem is to determine the paired-domination number, which is the minimum cardinality of a paired-dominating set. Motivated by a mistaken algorithm given by Chen, Kang and Ng [ Paired domination on interval and circular-arc graphs, Disc. Appl. Math. 155(2007Math. 155( ),2077Math. 155( -2086, we present two linear time algorithms to find a minimum cardinality paired-dominating set in block and interval graphs. In addition, we prove that paired-domination problem is NP-complete for bipartite graphs, chordal graphs, even split graphs.

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