A wheel is a graph formed by a chordless cycle C and a vertex u not in C that has at least three neighbors in C. We prove that every 3-connected planar graph that does not contain a wheel as an induced subgraph is either a line graph or has a clique cutset. We prove that every planar graph that does not contain a wheel as an induced subgraph is 3-colorable. AMS classification: 05C75 ∗Partially ...

In this paper we discuss cordial labeling in context of barycentric subdivision of shell graph, complete bipartite graph Kn,n and wheel graph. AMS subject classification: 05C78.

The celebrated grid exclusion theorem states that for every h-vertex planar graph H , there is a constant ch such that if a graph G does not contain H as a minor then G has treewidth at most ch. We are looking for patterns of H where this bound can become a low degree polynomial. We provide such bounds for the following parameterized graphs: the wheel (ch = O(h)), the double wheel (ch = O(h 2 ·...

In past years, topological indices are introduced to measure the characters of chemical molecules. Thus, the study of these topological indices has raised large attention in the field of chemical science, biology science and pharmaceutical science. In this paper, by virtue of molecular structure analysis, we determine the total Szeged index, vertex-edge Wiener index and edge hyper-Wiener index ...

A parallel minor is obtained from a graph by any sequence of edge contractions and parallel edge deletions. We prove that, for any positive integer k, every internally 4-connected graph of sufficiently high order contains a parallel minor isomorphic to a variation of K4,k with a complete graph on the vertices of degree k, the k-partition triple fan with a complete graph on the vertices of degre...

We show that the two problems of deciding whether k vertices or k edges can be deleted from a graph to obtain a wheel-free graph is W [2]-hard. This immediately implies that deciding whether k edges can be added to obtain a graph that contains no complement of a wheel as an induced subgraph isW [2]-hard, thereby resolving an open problem of Heggernes et. al. [7] (STOC07) who asks whether there ...

We introduce new classes of valid inequalities, called wheel inequalities, for the stable set polytope P G of a graph G. Each \wheel connguration" gives rise to two such inequalities. The simplest wheel connguration is an \odd" subdivision W of a wheel, and for these we give necessary and suucient conditions for the wheel inequality to be facet-inducing for P W. Generalizations arise by allowin...

We prove that, for every positive integer k, there is an integer N such that every 3-connected graph with at least N vertices has a minor isomorphic to the k-spoke wheel or K3,k; and that every internally 4-connected graph with at least N vertices has a minor isomorphic to the 2k-spoke double wheel, the k-rung circular ladder, the k-rung Möbius ladder, or K4,k. We also prove an analogous result...

There is a graph reduction system so that every optimal 1planar graph can be reduced to an irreducible extended wheel graph, provided the reductions are applied such that the given graph class is preserved. A graph is optimal 1-planar if it can be drawn in the plane with at most one crossing per edge and is optimal if it has the maximum of 4n− 8 edges. We show that the reduction system is conte...

Thtte's Wheels Theorem states that a minimally 3-connected non-wheel graph G with at least four vertices contains at least one edge e such that the contraction of e from G produces a graph which is both 3-connected and simple. The edge e is said to be non-essential. We show that a minimally 3-connected graph which is non-planar contains at least six non-essential edges. The wheel graphs are the...