Graph embedding with topological cycle-constraints?

Abstract

This paper concerns graph embedding under topological con-straints. We address the problem of finding a planar embedding of a graph satisfying a set of constraints between its vertices and cycles that require embedding a given vertex inside its corresponding cycle. This problem turns out to be NP-complete. However, towards an analysis of its tractable subproblems, we develop an efficient algorithm for the special case where graphs are 2-connected and any two distinct cycles in the constraints have at most one vertex in common.