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

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Hamiltonians
Hamiltonian path
Sparsity
Rigidity
Disjoint
Connectivity
Counterexample
Networks (circuits)
Graph in graph theory
Plane Graph
Simple Graph
Spanning tree
Maximum Degree
Undirected Graph
Connected graph
Denote
Sufficient Conditions

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

Cite this

Sparsity and connectivity of medial graphs : Concerning two edge-disjoint Hamiltonian paths in planar rigidity circuits. / Kijima, Shuji; Tanigawa, Shin Ichi.

In: Discrete Mathematics, Vol. 312, No. 16, 28.08.2012, p. 2466-2472.

Research output: Contribution to journalArticle

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