Removable cycles in non-bipartite graphs

K. Kawarabayashi , O. Lee , Bruce Reed (1)

1 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications

In this paper we prove the following result. Suppose that s and t are vertices of a 3-connected graph G such that G-s-t is not bipartite and there is no cutset X of size three in G for which some component U of G-X is disjoint from s,t. Then either (1) G contains an induced path P from s to t such that G-V(P) is not bipartite or (2) G can be embedded in the plane so that every odd face contains one of s or t. Furthermore, if (1) holds then we can insist that G-V(P) is connected, while if G is 5-connected then (1) must hold and P can be chosen so that G-V(P) is 2-connected.

Domaines: Informatique mobile; Réseaux et télécommunications [cs.NI]

K. Kawarabayashi, O. Lee, Bruce Reed. Removable cycles in non-bipartite graphs. Journal of Combinatorial Theory, Series B, 2009, 99, pp.30―38. ⟨10.1016/j.jctb.2008.03.007⟩. ⟨hal-00795287⟩