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Structures of Critical Nontree Graphs with Cutwidth Four

Author

Listed:
  • Zhenkun Zhang

    (School of Mathematics and Statistics, Huanghuai University, Zhumadian 463000, China)

  • Hongjian Lai

    (Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA)

Abstract

The cutwidth of a graph G is the smallest integer k ( k ≥ 1 ) such that the vertices of G are arranged in a linear layout [ v 1 , v 2 , . . . , v n ] , in such a way that for each i = 1 , 2 , . . . , n − 1 , there are at most k edges with one endpoint in { v 1 , v 2 , . . . , v i } and the other in { v i + 1 , . . . , v n } . The cutwidth problem for G is to determine the cutwidth k of G . A graph G with cutwidth k is k -cutwidth critical if every proper subgraph of G has a cutwidth less than k and G is homeomorphically minimal. In this paper, except five irregular graphs, other 4-cutwidth critical graphs were resonably classified into two classes, which are graph class with a central vertex v 0 , and graph class with a central cycle C q of length q ≤ 6 , respectively, and any member of two graph classes can skillfuly achieve a subgraph decomposition S with cardinality 2, 3 or 4, where each member of S is either a 2-cutwith graph or a 3-cutwidth graph.

Suggested Citation

  • Zhenkun Zhang & Hongjian Lai, 2023. "Structures of Critical Nontree Graphs with Cutwidth Four," Mathematics, MDPI, vol. 11(7), pages 1-22, March.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:7:p:1631-:d:1109398
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    References listed on IDEAS

    as
    1. Zhen-Kun Zhang & Hong-Jian Lai, 2017. "Characterizations of k-cutwidth critical trees," Journal of Combinatorial Optimization, Springer, vol. 34(1), pages 233-244, July.
    2. Zhen-Kun Zhang & Zhong Zhao & Liu-Yong Pang, 2022. "Decomposability of a class of k-cutwidth critical graphs," Journal of Combinatorial Optimization, Springer, vol. 43(2), pages 384-401, March.
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    1. Zhen-Kun Zhang & Zhong Zhao & Liu-Yong Pang, 2022. "Decomposability of a class of k-cutwidth critical graphs," Journal of Combinatorial Optimization, Springer, vol. 43(2), pages 384-401, March.

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