Complexity and algorithms for neighbor-sum-2-distinguishing {1,3}-edge-weighting of graphs

Abstract

A neighbor-sum-distinguishing W-edge-weighting of a graph G=(V,E) is an assignment ω of weights from a set of integers W to the set of edges E of G such that for every pair of adjacent vertices, the incident sums induced by the edge-weighting are different, where the incident sum of a vertex v induced by the edge-weighting ω is σ(v)=∑u∈N(v)ω(uv), where N(v) is the set of neighbors of v in G. Whereas, a neighbor-sum-2-distinguishing W-edge-weighting of G is a neighbor-sum-distinguishing W-edge-weighting such that the incident sums of every pair of adjacent vertices differ by at least 2. A recently proposed conjecture about neighbor-sum-2-distinguishing {1,3,5}-edge-weighting states that any graph with no component isomorphic to K2 admits a neighbor-sum-2-distinguishing {1,3,5}-edge-weighting. In this paper, we prove that deciding whether there exists a neighbor-sum-2-distinguishing {1,3}-edge-weighting is NP-complete for bipartite graphs. We present an algorithm that computes a neighbor-sum-2-distinguishing {1,3}-edge-weighting of the central graph of any graph in polynomial time and thus proves the above conjecture for the central graph of any graph. Further, we establish that if any pair of graphs G and H admit neighbor-sum-2-distinguishing edge-weightings, then so does their Cartesian product, G□H. We study another aspect of neighbor-sum-2-distinguishing edge-weighting of a graph G, namely, the minimum number of incident sums used by a neighbor-sum-2-distinguishing {1,3}-edge-weighting, which is denoted by γΣ>1,{1,3}(G). We prove that the problems of deciding whether there exists a neighbor-sum-2-distinguishing {1,3}-edge-weighting of G from a given set of sums, and deciding whether γΣ>1,{1,3}(G)≤k are NP-complete for bipartite graphs. On the positive side, we provide both upper and lower bounds on γΣ>1,{1,3}(C(G)) for central graphs of bipartite graphs and split graphs. We also propose an algorithm that computes the optimal neighbor-sum-2-distinguishing {1,3}-edge-weighting of the central graphs of cycles and paths in polynomial time.