On neighbour sum-distinguishing $\{0,1\}$-edge-weightings of bipartite graphs

TL;DR

This paper characterizes bipartite graphs and trees that can be distinguished by neighbor sums using only 0-1 edge weights, showing the problem is polynomial-time solvable for these classes but NP-complete in general.

Contribution

It provides a complete characterization of bipartite graphs and trees lacking the 0-1 property, establishing complexity boundaries for the neighbor sum-distinguishing problem.

Findings

Characterization of all bridgeless bipartite graphs without the 0-1 property

Characterization of all trees without the 0-1 property

The problem is in P for these classes but NP-complete for general graphs

Abstract

Let $S$ be a set of integers. A graph G is said to have the S-property if there exists an S-edge-weighting $w : E(G) \rightarrow S$ such that any two adjacent vertices have different sums of incident edge-weights. In this paper we characterise all bridgeless bipartite graphs and all trees without the $\{0,1\}$-property. In particular this problem belongs to P for these graphs while it is NP-complete for all graphs.