Related papers: Graphs with no induced wheel or antiwheel
One way to certify that a graph does not contain an induced cycle of length six is to provide a partition of its vertex set into (i) a stable set, and (ii) a graph containing no stable set of size three and no induced matching of size two.…
In a finite undirected simple graph, a chordless cycle is an induced subgraph which is a cycle. A graph is called cyclically orientable if it admits an orientation in which every chordless cycle is cyclically oriented. We propose an…
Let $W$ be any wheel graph and $\mathcal{G}$ the class of all countable graphs not containing $W$ as a minor. We show that there exists a graph in $\mathcal{G}$ which contains every graph in $\mathcal{G}$ as an induced subgraph.
A hole is a chordless cycle with at least four vertices. A hole is odd if it has an odd number of vertices. A dart is a graph which vertices $a, b, c, d, e$ and edges $ab, bc, bd, be, cd, de$. Dart-free graphs have been actively studied in…
Chordal graphs are the graphs in which every cycle of length at least four has a chord. A set $S$ is a vertex separator for vertices $a$ and $b$ if the removal of $S$ of the graph separates $a$ and $b$ into distinct connected components. A…
The sun is the graph obtained from a cycle of length even and at least six by adding edges to make the even-indexed vertices pairwise adjacent. Suns play an important role in the study of strongly chordal graphs. A graph is chordal if it…
What are the unavoidable induced subgraphs of graphs with large treewidth? It is well-known that the answer must include a complete graph, a complete bipartite graph, all subdivisions of a wall and line graphs of all subdivisions of a wall…
The complexity of testing whether a graph contains an induced odd cycle of length at least five is currently unknown. In this paper we show that this can be done in polynomial time if the input graph has no induced subgraph isomorphic to…
An odd wheel graph is a graph formed by connecting a new vertex to all vertices of an odd cycle. We answer a question of Rosenfeld and Le by showing that odd wheels cannot be drawn in the plane such that the lengths of the edges are odd…
A {\em hole} is a chordless cycle of length at least four. A hole is {\em even} (resp. {\em odd}) if it contains an even (resp. odd) number of vertices. A \emph{cap} is a graph induced by a hole with an additional vertex that is adjacent to…
We prove that a graph $G$ contains no induced $5$-vertex path and no induced complement of a $5$-vertex path if and only if $G$ is obtained from $5$-cycles and split graphs by repeatedly applying the following operations: substitution,…
A graph is "$\ell$-holed" if all its induced cycles of length at least four have length exactly $\ell$. We give a complete description of the $\ell$-holed graphs for each $\ell\ge 7$.
Given a graph $G$ and a graph property $P$ we say that $G$ is minimal with respect to $P$ if no proper induced subgraph of $G$ has the property $P$. An HC-obstruction is a minimal 2-connected non-Hamiltonian graph. Given a graph $H$, a…
A \emph{hole} in a graph is an induced cycle with at least 4 vertices. A graph is \emph{even-hole-free} if it does not contain a hole on an even number of vertices. A \emph{pyramid} is a graph made of three chordless paths $P_1 = a \dots…
A wheel graph consists of a cycle along with a center vertex connected to every vertex in the cycle. In this paper we show that every subgraph of a wheel graph has list coupled chromatic number at most 5, and this coloring can be found in…
A hole in a graph is an induced subgraph which is a cycle of length at least four. We prove that for every positive integer k, every triangle-free graph with sufficiently large chromatic number contains holes of k consecutive lengths.
We determine, for all $k\geq 6$, the typical structure of graphs that do not contain an induced $2k$-cycle. This verifies a conjecture of Balogh and Butterfield. Surprisingly, the typical structure of such graphs is richer than that…
Let us say a graph is $s\mathcal{O}$-free, where $s\ge 1$ is an integer, if there do not exist $s$ cycles of the graph that are pairwise vertex-disjoint and have no edges joining them. The structure of such graphs, even when $s=2$, is not…
We consider the class of graphs containing no odd hole, no odd antihole, and no configuration consisting of three paths between two nodes such that any two of the paths induce a hole, and at least two of the paths are of length 2. This…
A graph is unichord free if it does not contain a cycle with exactly one chord as its subgraph. In [3], it is shown that a graph is unichord free if and only if every minimal vertex separator is a stable set. In this paper, we first show…