Forbidden pairs of disconnected graphs for supereulerianity of connected graphs

Abstract

Let H be a set of graphs. A graph G is said to be H-free if G does not contain H as an induced subgraph for all H∈H, and we call H a forbidden pair if |H|=2. A graph is called supereulerian if it contains a spanning connected even subgraph. In 1979, Pulleyblank showed that determining whether a graph is supereulerian, even when restricted to planar graphs, is NP-complete. In this paper, we characterize all pairs of graphs R,S (not necessary connected) such that every 2-connected or 2-edge-connected {R,S}-free graph (of sufficiently large order) is supereulerian. We also characterize all minimal 2-connected non-supereulerian graphs, which extends a result by the third author and Xiong (2017) [29].