Some forbidden subgraph conditions for a graph to have a k-contractible edge

AbstractMotivated by several conjectures due to Nikoghosyan, in a recent article due to Li et al., the aim was to characterize all possible graphs H such that every 1-tough H-free graph is hamiltonian. The almost complete answer was given there by the conclusion that every proper induced subgraph H of $$K_1\cup P_4$$ K 1 ∪ P 4 can act as a forbidden subgraph to ensure that every 1-tough H-free graph is hamiltonian, and that there is no other forbidden subgraph with this property, except possibly for the graph $$K_1\cup P_4$$ K 1 ∪ P 4 itself. The hamiltonicity of 1-tough $$K_1\cup P_4$$ K 1 ∪ P 4 -free graphs, as conjectured by Nikoghosyan, was left there as an open case. In this paper, we consider the stronger property of pancyclicity under the same condition. We find that the results are completely analogous to the hamiltonian case: every graph H such that any 1-tough H-free graph is hamiltonian also ensures that every 1-tough H-free graph is pancyclic, except for a few specific classes of graphs. Moreover, there is no other forbidden subgraph having this property. With respect to the open case for hamiltonicity of 1-tough $$K_1\cup P_4$$ K 1 ∪ P 4 -free graphs we give infinite families of graphs that are not pancyclic.

Download Full-text

Forbidden ordered subgraph vs. forbidden subgraph characterizations of graph classes

Journal of Graph Theory ◽

10.1002/(sici)1097-0118(199902)30:2<71::aid-jgt1>3.0.co;2-g ◽

1999 ◽

Vol 30 (2) ◽

pp. 71-76 ◽

Cited By ~ 1

Author(s):

Mark Ginn

Keyword(s):

Forbidden Subgraph ◽

Graph Classes

Download Full-text

Matrix Partitions with Finitely Many Obstructions

The Electronic Journal of Combinatorics ◽

10.37236/976 ◽

2007 ◽

Vol 14 (1) ◽

Cited By ~ 8

Author(s):

Tomás Feder ◽

Pavol Hell ◽

Wing Xie

Keyword(s):

Symmetric Matrix ◽

Forbidden Subgraph ◽

Input Graph ◽

Induced Subgraphs ◽

Induced Subgraph ◽

Forbidden Induced Subgraphs ◽

Extra Effort ◽

Matrix Partition ◽

Finite Set ◽

Forbidden Induced Subgraph

Each $m$ by $m$ symmetric matrix $M$ over $0, 1, *$, defines a partition problem, in which an input graph $G$ is to be partitioned into $m$ parts with adjacencies governed by $M$, in the sense that two distinct vertices in (possibly equal) parts $i$ and $j$ are adjacent if $M(i,j)=1$, and nonadjacent if $M(i,j)=0$. (The entry $*$ implies no restriction.) We ask which matrix partition problems admit a characterization by a finite set of forbidden induced subgraphs. We prove that matrices containing a certain two by two diagonal submatrix $S$ never have such characterizations. We then develop a recursive technique that allows us (with some extra effort) to verify that matrices without $S$ of size five or less always have a finite forbidden induced subgraph characterization. However, we exhibit a six by six matrix without $S$ which cannot be characterized by finitely many induced subgraphs. We also explore the connection between finite forbidden subgraph characterizations and related questions on the descriptive and computational complexity of matrix partition problems.

Download Full-text