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Essentially 3-edge-connected reduced graph of diameter three

Author

Listed:
  • Ye, Fulong
  • Qin, Xiaoxiao
  • Zhan, Mingquan
  • Huo, Bofeng

Abstract

A graph is called supereulerian if it possesses a spanning connected subgraph such that each of its vertices is of even degree. In order to explore the structure of supereulerian graphs, Catlin formulated the definitions of collapsible graphs and reduced graphs in 1988. The significance of reduced graphs stems from a theorem by Catlin, which establishes that a graph is supereulerian precisely when its reduced graph is supereulerian. In 1990, Lai determined all reduced graphs with diameter 2. We resolve the problem of characterizing reduced graphs with essential edge connectivity exactly 3, diameter 3, and minimum degree at least 3, and conclude that G=P(14) is the unique graph with these conditions. Furthermore, we conjecture that if G is a 3-edge-connected reduced graph with diameter 3, then G must belong to one of four well-characterized exceptional graph families.

Suggested Citation

  • Ye, Fulong & Qin, Xiaoxiao & Zhan, Mingquan & Huo, Bofeng, 2026. "Essentially 3-edge-connected reduced graph of diameter three," Applied Mathematics and Computation, Elsevier, vol. 527(C).
    • Handle: RePEc:eee:apmaco:v:527:y:2026:i:c:s0096300326001591
    DOI: 10.1016/j.amc.2026.130107
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