A simple undirected graph G=(V,E) is a rigidity circuit if |E|=2|V|-2 and |E G[X]|≤2|X|-3 for every X⊂V with 2≤|X|≤|V|-1, where E G[X] denotes the set of edges connecting vertices in X. It is known that a rigidity circuit can be decomposed into two edge-disjoint spanning trees. Graver et al. (1993)  asked if any rigidity circuit with maximum degree 4 can be decomposed into two edge-disjoint Hamiltonian paths. This paper presents infinitely many counterexamples for the question. Counterexamples are constructed based on a new characterization of a 3-connected plane graph in terms of the sparsity of its medial graph and a sufficient condition for the connectivity of medial graphs.
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics