For several graph theoretic parameters such as vertex cover and dominating set, it is known that if their values are bounded by k then the treewidth of the graph is bounded by some function of k. This fact is used as the main tool for the design of several fixed-parameter algorithms on minor-closed graph classes such as planar graphs, single-crossing-minor-free graphs, and graphs of bounded gen...

We prove upper and lower bounds on the size of the largest square grid graph that is a subgraph, minor, or shallow minor of a graph in the form of a larger square grid from which a specified number of vertices have been deleted. Our bounds are tight to within constant factors. We also provide less-tight bounds on analogous problems for higher-dimensional grids.

We give the first linear kernels for the Dominating Set and Connected Dominating Set problems on graphs excluding a fixed graph H as a topological minor. In other words, we prove the existence of polynomial time algorithms that, for a given H-topological-minor-free graph G and a positive integer k, output an H-topologicalminor-free graph G′ on O(k) vertices such that G has a (connected) dominat...

We prove that for any fixed r ≥ 2, the tree-width of graphs not containing Kr as a topological minor (resp. as a subgraph) is bounded by a linear (resp. polynomial) function of their rank-width. We also present refinements of our bounds for other graph classes such as Kr-minor free graphs and graphs of bounded genus.

‎let $r$ be a commutative ring with non-zero identity. ‎we describe all $c_3$‎- ‎and $c_4$-free intersection graph of non-trivial ideals of $r$ as well as $c_n$-free intersection graph when $r$ is a reduced ring. ‎also, ‎we shall describe all complete, ‎regular and $n$-claw-free intersection graphs. ‎finally, ‎we shall prove that almost all artin rings $r$ have hamiltonian intersection graphs. ...

The Kit ligand (KL)/Kit receptor pair functions in hematopoiesis, gametogenesis, and melanogenesis. KL is encoded at the murine steel (Sl) locus and encodes a membrane growth factor which may be proteolytically processed to produce soluble KL. The membrane-associated form of KL is critical in mediating Kit function in vivo. Evidence for a role of cytoplasmic domain sequences of KL comes from th...

Tree-width and its linear variant path-width play a central role for the graph minor relation. In particular, Robertson Seymour (1983) proved that every tree~$T$, class of graphs do not contain $T$ as has bounded path-width. For pivot-minor relation, rank-width take over from tree-width As such, it is natural to examine if rank-width. We first prove this statement false whenever tree caterpilla...

The class of graphs with no K3,t-minors, t ≥ 3, contains all planar graphs and plays an important role in graph minor theory. In 1992, Seymour and Thomas conjectured the existence of a function α(t) > 0 and a constant β > 0, such that every 3-connected n-vertex graph with no K3,t-minors, t ≥ 3, contains a cycle of length at least α(t)n . The purpose of this paper is to confirm this conjecture w...

We construct an in nite planar graph that contains every planar graph as a minor.

A cubic graph G is uniquely edge-3-colorable if G has precisely one 1-factorization. It is proved in this paper, if a uniquely edge-3-colorable, cubic graph G is cyclically 4-edgeconnected, but not cyclically 5-edge-connected, then G must contain a snark as a minor. This is an approach to a conjecture that every triangle free uniquely edge-3-colorable cubic graph must have the Petersen graph as...