Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4649784 | Discrete Mathematics | 2009 | 4 Pages |
Abstract
Let τ(G)τ(G) denote the number of vertices in a longest path in a graph G=(V,E)G=(V,E). A subset KK of VV is called a PnPn-kernel of GG if τ(G[K])≤n−1τ(G[K])≤n−1 and every vertex v∈V∖Kv∈V∖K is adjacent to an end-vertex of a path of order n−1n−1 in G[K]G[K]. It is known that every graph has a PnPn-kernel for every positive integer n≤9n≤9. R. Aldred and C. Thomassen in [R.E.L. Aldred, C. Thomassen, Graphs with not all possible path-kernels, Discrete Math. 285 (2004) 297–300] proved that there exists a graph which contains no P364P364-kernel. In this paper, we generalise this result. We construct a graph with no P155P155-kernel and for each integer l≥0l≥0 we provide a construction of a graph GG containing no Pτ(G)−lPτ(G)−l-kernel.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Discrete Mathematics and Combinatorics
Authors
Peter Katrenič, Gabriel Semanišin,