Sparsity and connectivity of medial graphs: Concerning two edge-disjoint Hamiltonian paths in planar rigidity circuits

Shuji Kijima, Shin Ichi Tanigawa

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

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) [5] 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.

Original languageEnglish
Pages (from-to)2466-2472
Number of pages7
JournalDiscrete Mathematics
Volume312
Issue number16
DOIs
Publication statusPublished - Aug 28 2012

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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