Related papers: Apex Graphs and Cographs

The family of graphs that can be constructed from isolated vertices by disjoint union and graph join operations are called cographs. These graphs can be represented in a tree-like representation termed parse tree or cotree. In this paper,…

We establish tight lower and upper bounds on the number of edges in traceable graphs in several classes of dense graphs. A graph is traceable if it has a Hamiltonian path. We show that the bound is: - quadratic for the class of graphs of…

The enhanced power graph of a group is the simple graph whose vertex set is consisted of all elements of the group, and whose any pair of vertices are adjacent if they generate a cyclic subgroup. In this paper, we classify all finite groups…

We discover new hereditary classes of graphs that are minimal (with respect to set inclusion) of unbounded clique-width. The new examples include split permutation graphs and bichain graphs. Each of these classes is characterised by a…

Cographs are graphs in which no four vertices induce a simple connected path $P_4$. Cograph editing is to find for a given graph $G = (V,E)$ a set of at most $k$ edge additions and deletions that transform $G$ into a cograph. This…

A graph $\textit{G}$ is a tuple $(\textit{V}, \textit{E})$, where $\textit{V}$ is the vertex set, $\textit{E}$ is the edge set. A reduced graph is a graph of deleting non-Hamiltonian edges and smoothing out the redundant vertices of degree…

Given two graphs G and H, we ask under which conditions there is a relation R that generates the edges of H given the structure of graph G. This construction can be seen as a form of multihomomorphism. It generalizes surjective…

A graph $G$ is a {\em chordal-$k$-generalized split graph} if $G$ is chordal and there is a clique $Q$ in $G$ such that every connected component in $G[V \setminus Q]$ has at most $k$ vertices. Thus, chordal-$1$-generalized split graphs are…

We prove that a hereditary class of graphs is $(\mathsf{tw}, \omega)$-bounded if and only if the induced minors of the graphs from the class form a $(\mathsf{tw}, \omega)$-bounded class.

Word-representable graphs, characterized by the existence of a semi-transitive orientation, form a well-studied class of graphs. Comparability graphs form another well-studied class and constitute a subclass of word-representable graphs.…

A class of graphs is $\chi$-bounded if there is a function $f$ such that every graph $G$ in the class has chromatic number at most $f(\omega(G))$, where $\omega(G)$ is the clique number of $G$; the class is polynomially $\chi$-bounded if…

We first prove that for every vertex x of a 4-connected graph G there exists a subgraph H in G isomorphic to a subdivision of the complete graph K4 on four vertices such that G-V(H) is connected and contains x. This implies an affirmative…

We prove for every graph H there exists a>0 such that, for every graph G with at least two vertices, if no induced subgraph of G is a subdivision of H, then either some vertex of G has at least a|G| neighbours, or there are two disjoint…

A graph $G$ is said to be an $(s, k)$-polar graph if its vertex set admits a partition $(A, B)$ such that $A$ and $B$ induce, respectively, a complete $s$-partite graph and the disjoint union of at most $k$ complete graphs. Polar graphs and…

A k-ranking of a graph G is a labeling of the vertices of G with values from {1,...,k} such that any path joining two vertices with the same label contains a vertex having a higher label. The tree-depth of G is the smallest value of k for…

Transmission of a vertex v of a connected graph G is the sum of distances from v to all other vertices in G. Graph G is transmission irregular (TI) if no two of its vertices have the same transmission, and G is interval transmission…

A graph $G$ is said to be the intersection of graphs $G_1,G_2,\ldots,G_k$ if $V(G)=V(G_1)=V(G_2)=\cdots=V(G_k)$ and $E(G)=E(G_1)\cap E(G_2)\cap\cdots\cap E(G_k)$. For a graph $G$, $\mathrm{dim}_{COG}(G)$ (resp. $\mathrm{dim}_{TH}(G)$)…

In this paper, we characterize the class of {\em contraction perfect} graphs which are the graphs that remain perfect after the contraction of any edge set. We prove that a graph is contraction perfect if and only if it is perfect and the…

A graph $G$ is a $B_0$-VPG graph if one can associate a path on a rectangular grid with each vertex such that two vertices are adjacent if and only if the corresponding paths intersect at at least one grid-point. A graph $G$ is a contact…

We study the complexity of the Graph Isomorphism problem on graph classes that are characterized by a finite number of forbidden induced subgraphs, focusing mostly on the case of two forbidden subgraphs. We show hardness results and develop…