Structures of Critical Nontree Graphs with Cutwidth Four

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 [v1,v2,...,vn], in such a way that for each i=1,2,...,n−1, there are at most k edges with one endpoint in {v1,v2,...,vi} and the other in {vi+1,...,vn}. 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 v0, and graph class with a central cycle Cq 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.