Decompositions of critical trees with cutwidth k

  • Zhen-Kun ZhangEmail author


The cutwidth of a graph G is the minimum integer k such that the vertices of G are arranged in a linear layout \([v_1,\ldots ,v_n]\) in such a way that, for every \(j= 1,\ldots ,n-1\), there are at most k edges with one endpoint in \(\{v_1,\ldots ,v_j\}\) and the other in \(\{v_{j+1},\ldots ,v_n\}\). 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 cutwidth less than k and G is homeomorphically minimal. In this paper, we obtain that any k-cutwidth critical tree \(\mathcal {T}\) can be decomposed into three \((k-1)\)-cutwidth critical subtrees for \(k\ge 2\); And an \(O(|V(\mathcal {T})|^{2}\mathrm{log} |V(\mathcal {T})|)\) algorithm of computing the three subtrees is given.


Graph labeling Cutwidth Critical tree Decomposition 

Mathematics Subject Classification

05C75 05C78 90C27 



The author would like to thank the referees for their helpful suggestions on improving the representation of this paper.


  1. Bondy JA, Murty USR (2008) Graph theory. Springer, NewYorkCrossRefGoogle Scholar
  2. Chung FRK, Seymour PD (1985) Graphs with small bandwidth and cutwidth. Discret Math 75:268–277MathSciNetGoogle Scholar
  3. Chung MJ, Makedon F, Sudborough IH, Turner J (1985) Polynomial time algorithms for the min cut problem on degree restricted trees. SIAM J Comput 14:158–177MathSciNetCrossRefGoogle Scholar
  4. Diaz J, Petit J, Serna M (2002) A survey of graph layout problems. ACM Comput Surv 34:313–356CrossRefGoogle Scholar
  5. Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. W.H. Freeman & Company, San FranciscozbMATHGoogle Scholar
  6. Korach E, Solel N (1993) Tree-width, path-width and cutwidth. Discret Appl Math 43:97–101MathSciNetCrossRefGoogle Scholar
  7. Lin Y, Yang A (2004) On 3-cutwidth critical graphs. Discret Math 275:339–346MathSciNetCrossRefGoogle Scholar
  8. Thilikos DM, Serna M, Bodlaender HL (2005) Cutwidth II: algorithms for partial w-trees of bounded degree. J Algorithms 56(1):25–49MathSciNetCrossRefGoogle Scholar
  9. Yannakakis M (1985) A polynomial algorithm for the min-cut linear arrangement of trees. J ACM 32:950–988MathSciNetCrossRefGoogle Scholar
  10. Zhang Z, Lin Y (2012) On 4-cutwidth critical trees. Ars Comb 105:149–160MathSciNetzbMATHGoogle Scholar
  11. Zhang Z, Lai H (2017) Characterizations of k-cutwidth critical trees. J Comb Optim 34:233–244MathSciNetCrossRefGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  1. 1.Huanghuai UniversityZhumadianChina

Personalised recommendations