Symmetric core and spanning trails in directed networks

Abstract

A digraph D is supereulerian if D has a spanning closed trail, and is strongly trail-connected if for any pair of vertices u,v∈V(D), D has a spanning (u,v)-trail and a spanning (v,u)-trail. The symmetric core J=J(D) of a digraph D is a spanning subdigraph of D with A(J) consisting of all symmetric arcs in D. Let J1,J2,⋯,Jk(D) be the connected symmetric components of J and define λ0(D)=min⁡{λ(Ji):1≤i≤k(D)}. We prove that the contraction D′=D/J can be used to predict the existence of spanning trails in D. It is known that if k(D)≤2, then D has a spanning closed trail. In particular, each of the following holds for a strong digraph D with k(D)≥3. (i) If λ0(D)≥k(D)−2, then D has a spanning trail if and only if D′ has a spanning trail. (ii) If λ0(D)≥k(D)−1, then D is supereulerian if and only if D′ is supereulerian. (iii) If λ0(D)≥k(D), then D is strongly trail-connected if and only if D′ is strongly trail-connected.