Related papers: Using $p$-row graphs to study $p$-competition grap…

Given a graph $G$, a subgraph $H$ is isometric if $d_H(u,v) = d_G(u,v)$ for every pair $u,v\in V(H)$, where $d$ is the distance function. A graph $G$ is distance preserving (dp) if it has an isometric subgraph of every possible order. A…

Let $G=(V,E)$ be a graph and $p$ a positive integer. A subset $S\subseteq V$ is called a $p$-dominating set of $G$ if every vertex not in $S$ has at least $p$ neighbors in $S$. The $p$-domination number is the minimum cardinality of a…

The niche graph of a digraph $D$ is the (simple undirected) graph which has the same vertex set as $D$ and has an edge between two distinct vertices $x$ and $y$ if and only if $N^+_D(x) \cap N^+_D(y) \neq \emptyset$ or $N^-_D(x) \cap…

An ordered graph is a graph enhanced with a linear order on the vertex set. An ordered graph is a core if it does not have an order-preserving homomorphism to a proper subgraph. We say that $H$ is the core of $G$ if (i) $H$ is a core, (ii)…

In this paper, we introduce the notion of the containment graph of a family of sets and containment classes of graphs and posets. Let $Z$ be a family of nonempty sets. We call a (simple, finite) graph G = (V, E) a $Z$-containment graph…

We say that a digraph $D$ is $(i,j)$-step competitive if any two vertices have an $(i,j)$-step common out-neighbor in $D$ and that a graph $G$ is $(i,j)$-step competitively orientable if there exists an $(i,j)$-step competitive orientation…

The $k$-dominating graph $D_k(G)$ of a graph $G$ is defined on the vertex set consisting of dominating sets of $G$ with cardinality at most $k$, two such sets being adjacent if they differ by either adding or deleting a single vertex. A…

The competition graph of a doubly partial order is known to be an interval graph. The CCE graph and the niche graph of a doubly partial order are also known to be interval graphs if the graphs do not contain a cycle of length four and three…

Given a graph $G$ then a subgraph $H$ is $isometric$ if, for every pair of vertices $u,v$ of $H$, we have $d_H(u,v) = d_G(u,v)$. We say a graph $G$ is $distance\ preserving\ (dp)$ if it has an isometric subgraph of every possible order up…

If each vertex of an acyclic digraph has indegree at most $i$ and outdegree at most $j$, then it is called an $(i,j)$ digraph, which was introduced by Hefner~{\it et al.}~(1991). Whereas Hefner~{\it et al.} characterized $(i,j)$ digraphs…

Let $\mathcal{D}$ be a set of straight-line segments in the plane, potentially crossing, and let $c$ be a positive integer. We denote by $P$ the union of the endpoints of the straight-line segments of $\mathcal{D}$ and of the intersection…

Given a graph G=(V, E), a vertex is said to ve-dominate an edge if it is either incident with the edge or adjacent to one of its endpoints. A set of vertices is a ve-dominating set if it ve-dominates every edge of the graph. We introduce…

We consider the following generalization of dominating sets: Let $G$ be a host graph and $P$ be a pattern graph $P$. A dominating $P$-pattern in $G$ is a subset $S$ of vertices in $G$ that (1) forms a dominating set in $G$ \emph{and} (2)…

Since Cho and Kim (2005) showed that the competition graph of a doubly partial order is an interval graph, it has been actively studied whether or not the same phenomenon occurs for other variants of competition graph and interesting…

Competition graphs were created in connected to a biological model as a means of reflecting the competition relations among the predators in the food webs and determining the smallest dimension of ecological phase space. In 2011, Factor and…

The median of a graph $G$ with weighted vertices is the set of all vertices $x$ minimizing the sum of weighted distances from $x$ to the vertices of $G$. For any integer $p\ge 2$, we characterize the graphs in which, with respect to any…

Let $G$ be a graph, a dominating induced matching (DIM) of $G$ is an induced matching that dominates every edge of $G$. In this paper we show that if a graph $G$ has a DIM, then $\chi(G) \leqslant 3$. Also, it is shown that if $G$ is a…

The graph G(p) associated with the p-groups of maximal class is a major tool in their classification. We introduce a subgraph of the graph G(p) called its frame. Its construction is based on the Lazard correspondence. We show that every…

A graph $X$ is said to be a pattern polynomial graph if its adjacency algebra is a coherent algebra. In this study we will find a necessary and sufficient condition for a graph to be a pattern polynomial graph. Some of the properties of the…

An ordered graph $G$ is a graph whose vertex set is a subset of integers. The edges are interpreted as tuples $(u,v)$ with $u < v$. For a positive integer $s$, a matrix $M \in \mathbb{Z}^{s \times 4}$, and a vector $\mathbf{p} =…