Cycles having the same modularity and removable edges in 2-connected graphs

  • Authors:

  • Kiyoshi Ando;Mariko Hagita;Atsushi Kaneko;Mikio Kano;Ken-ichi Kawarabayashi;Akira Saito

  • Affiliations:

  • Department of Information and Communication Engineering, The University of Electro-Communications, Chofu, Tokyo 182-8585, Japan;Faculty of Environmental Information, Keio University, Endo 5322, Fujisawa, Kanagawa 252-8520, Japan;Department of Computer Science and Communication Engineering, Kogakuin University, Nishi-Shinjuku 1-24-2, Shinjuku-Ku, Tokyo 163-8677, Japan;Department of Computer and Information Sciences, Ibaraki University, Nakanarusawa 4-12-1, Hitachi, Ibaraki 316-8511, Japan;Department of Mathematics, Keio University, Hiyoshi 3-14-1, Kohoku-Ku, Yokohama 223-8522, Japan;Department of Applied Mathematics, Nihon University, Sakurajosui 3-25-40, Setagaya-Ku, Tokyo 156-8550, Japan

  • Venue:
  • Discrete Mathematics
  • Year:
  • 2003

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Abstract

In this paper, we consider 2-connected multigraphs in which every cycle has length congruent to a modulo b (b ≥ 2). We prove that there exists such a multigraph which is homormorphic to a graph with minimum degree at least three only if a = 0, and that there exists such a graph only if a = 0 and b = 2. We also study the distribution of paths whose internal vertices have degree exactly two, and show a relation between these paths and edges in a 2-connected graph whose deletion results in a 2-connected graph.