Minimum spanning tree cycle intersection problem on outerplanar graphs

Abstract

We study the minimum spanning tree cycle intersection (MSTCI) problem on outerplanar graphs in this paper. Consider a connected simple graph G=(V,E) and any spanning tree T=(V,ET) of G, it is well-known that each non-tree edge e∈E∖ET induces a fundamental cycle in T∪{e}. We say that two distinct fundamental cycles intersect if they share at least one edge. The MSTCI problem seeks a spanning tree such that the number of pairwise cycle intersections is minimized. We provide a linear-time algorithm to solve the problem on outerplanar graphs by exploiting the cycle-cut duality on plane graphs. Besides, we show that the minimum cycle intersection number is bounded by O(n) for outerplanar graphs with n vertices.