Related papers: On strongly chordal graphs that are not leaf power…

A graph $G$ is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function $w$ is defined on its vertices. Then $G$ is $w$-well-covered if all maximal independent sets are of the same weight.…

A hereditary class H of graphs is $\chi$-bounded if there is a $\chi$-binding function f such that for every $G$ in $H$, $\chi(G)$ less than or equal to $f(\omega(G))$. Here we prove that if a graph $G$ is free of 1. {Chair; P$_4$+K$_1$} or…

The enhanced power graph of a finite group $G$, denoted by $\mathcal{P}_E(G)$, is the simple undirected graph whose vertex set is $G$ and two distinct vertices $x, y$ are adjacent if $x, y \in \langle z \rangle$ for some $z \in G$. In this…

The goal of this work is to give precise bounds on the counting complexity of a family of generalized coloring problems (list homomorphisms) on bounded-treewidth graphs. Given graphs $G$, $H$, and lists $L(v)\subseteq V(H)$ for every $v\in…

Tree-based phylogenetic networks, which may be roughly defined as leaf-labeled networks built by adding arcs only between the original tree edges, have elegant properties for modeling evolutionary histories. We answer an open question of…

A path in an(a) edge(vertex)-colored graph is called \emph{a conflict-free path} if there exists a color used on only one of its edges(vertices). An(A) edge(vertex)-colored graph is called \emph{conflict-free (vertex-)connected} if there is…

A numbering $f$ of a graph $G$ of order $n$ is a labeling that assigns distinct elements of the set $\{1,2, \ldots, n \}$ to the vertices of $G$. The strength $\mathrm{str}\left(G\right) $ of $G$ is defined by $\mathrm{str}\left( G\right)…

For integer $k\geq2,$ a graph $G$ is called $k$-leaf-connected if $|V(G)|\geq k+1$ and given any subset $S\subseteq V(G)$ with $|S|=k,$ $G$ always has a spanning tree $T$ such that $S$ is precisely the set of leaves of $T.$ Thus a graph is…

A family of graphs $\mathcal{F}$ is said to have the joint embedding property (JEP) if for every $G_1, G_2\in \mathcal{F}$, there is an $H\in \mathcal{F}$ that contains both $G_1$ and $G_2$ as induced subgraphs. If $\mathcal{F}$ is given by…

Path graphs are intersection graphs of paths in a tree.~In this paper we give a "6\ good characterization" of path graphs, namely, we prove that path graph membership is in $NP\cap CoNP$ without resorting to existing polynomial time…

The power graph P(G) of a group G is a graph with vertex set G, where two vertices u and v are adjacent if and only if one is the power of the other. In this paper, we raise and study the following question: For which natural numbers n…

We present an algorithm for determining whether a bipartite graph $G$ is 2-chordal (formerly doubly chordal bipartite). At its core this algorithm is an extension of the existing efficient algorithm for determining whether a graph is…

In this paper we extend the theory of bidimensionality to two families of graphs that do not exclude fixed minors: map graphs and power graphs. In both cases we prove a polynomial relation between the treewidth of a graph in the family and…

For a fixed set ${\cal H}$ of graphs, a graph $G$ is ${\cal H}$-subgraph-free if $G$ does not contain any $H \in {\cal H}$ as a (not necessarily induced) subgraph. A recently proposed framework gives a complete classification on ${\cal…

A graph $G$ is said to be a `set graph' if it admits an acyclic orientation that is also `extensional', in the sense that the out-neighborhoods of its vertices are pairwise distinct. Equivalently, a set graph is the underlying graph of the…

The power graph $P(G)$ of a finite group $G$ is the undirected simple graph with vertex set $G$, where two elements are adjacent if one is a power of the other. In this paper, the matching numbers of power graphs of finite groups are…

The perfect phylogeny problem is a classic problem in computational biology, where we seek an unrooted phylogeny that is compatible with a set of qualitative characters. Such a tree exists precisely when an intersection graph associated…

A graph $G$ is $1$-extendible if every edge belongs to at least one $1$-factor of $G$. Let $G$ be a graph with a $1$-factor $F$. Then an even $F$-orientation of $G$ is an orientation in which each $F$-alternating cycle has exactly an even…

The enhanced power graph of a group $G$ is a graph with vertex set $G,$ where two distinct vertices $x$ and $y$ are adjacent if and only if there exists an element $w$ in $G$ such that both $x$ and $y$ are powers of $w.$ In this paper, we…

Let $G$ be an undirected simple connected graph. We say a vertex $u$ is eccentric to a vertex $v$ in $G$ if $d(u,v)=\max\{d(v,w): w\in V(G)\}$. The eccentric graph, $E(G)$ of $G$ is a graph defined on the same vertex set as of $G$ and two…