Related papers: Generalizing Cographs to 2-Cographs
Let $P$ be a graph with a vertex $v$ such that $P\backslash v$ is a forest, and let $Q$ be an outerplanar graph. We prove that there exists a number $p=p(P,Q)$ such that every 2-connected graph of path-width at least $p$ has a minor…
For each integer $n\geq 3$, let $W_n$ denote the wheel graph obtained by connecting a single vertex to all vertices of a cycle of length $n$. In particular, $W_6$ is obtained from the Petersen graph by contracting three edges incident with…
A k-digraph is an orientation of a multi-graph that is without loops and contains at most k edges between any pair of distinct vertices. We obtain necessary and sufficient conditions for a sequence of non-negative integers in non-decreasing…
The celebrated Erd\H{o}s-P\'{o}sa Theorem, in one formulation, asserts that for every $c\geq 1$, graphs with no subgraph (or equivalently, minor) isomorphic to the disjoint union of $c$ cycles have bounded treewidth. What can we say about…
A spanning subgraph of a graph G is called a [0,2]-factor of G, if for . is a union of some disjoint cycles, paths and isolate vertices, that span the graph G. It is easy to get a [0,2]-factor of G and there would be many of [0,2]-factors…
A {\em hole} in a graph is an induced subgraph which is a cycle of length at least four. A hole is called {\em even} if it has an even number of vertices. An {\em even-hole-free} graph is a graph with no even holes. A vertex of a graph is…
The Johnson graph J(n,N) is defined as the graph whose vertices are the n-subsets of the set {1,2,...,N}, where two vertices are adjacent if they share exactly n - 1 elements. Unlike Johnson graphs, induced subgraphs of Johnson graphs (JIS…
We exhibit several families of planar graphs that are minor-minimal intrinsically spherical $3$-linked. A graph is intrinsically spherical 3-linked if it is planar graph that has, in every spherical embedding, a non-split 3-link consisting…
A graph is called a nut graph if zero is its eigenvalue of multiplicity one and its corresponding eigenvector has no zero entries. A graph is a bicirculant if it admits an automorphism with two equally sized vertex orbits. There are four…
Abstract inner automorphisms can be used to promote any category into a 2-category, and we study two-dimensional limits and colimits in the resulting 2-categories. Existing connected colimits and limits in the starting category become…
A graph is chordal if every induced cycle has three vertices. The Hadwiger number is the order of the largest complete minor of a graph. We characterize the chordal graphs in terms of the Hadwiger number and we also characterize the…
A directed graph $G=(V,E)$ is strongly biconnected if $G$ is strongly connected and its underlying graph is biconnected. A strongly biconnected directed graph $G=(V,E)$ is called $2$-vertex-strongly biconnected if $|V|\geq 3$ and the…
A 2-switch is an edge addition/deletion operation that changes adjacencies in the graph while preserving the degree of each vertex. A well known result states that graphs with the same degree sequence may be changed into each other via…
A 2-group is a "categorified" version of a group, in which the underlying set G has been replaced by a category and the multiplication map has been replaced by a functor. Various versions of this notion have already been explored; our goal…
In this paper, we introduce the concept of $k$-integral graphs. A graph $\Gamma$ is called $k$-integral if the extension degree of the splitting field of the characteristic polynomial of $\Gamma$ over rational field $\mathbb Q$ is equal to…
This paper studies induced paths in strongly regular graphs. We give an elementary proof that a strongly regular graph contains a path $P_4$ as an induced subgraph if and only if it is primitive, i.e. it is neither a complete multipartite…
Conduction graphs are defined here in order to elucidate at a glance the often complicated conduction behaviour of molecular graphs as ballistic molecular conductors. The graph $G^{\mathrm C}$ describes all possible conducting devices…
In this paper, our goal is to characterize two graph classes based on the properties of minimal vertex (edge) separators. We first present a structural characterization of graphs in which every minimal vertex separator is a stable set. We…
We prove that every graph $G$ on $n$ vertices with no isolated vertices contains an induced subgraph of size at least $n/10000$ with all degrees odd. This solves an old and well-known conjecture in graph theory.
For a given graph $R$, a graph $G$ is $R$-free if $G$ does not contain $R$ as an induced subgraph. It is known that every $2$-tough graph with at least three vertices has a $2$-factor. In graphs with restricted structures, it was shown that…