Related papers: Characterizing [h,2,1] graphs by minimal forbidden…

A path in a vertex-colored graph is called \emph{vertex-rainbow} if its internal vertices have pairwise distinct colors. A graph $G$ is \emph{rainbow vertex-connected} if for any two distinct vertices of $G$, there is a vertex-rainbow path…

A tree $T$ in an edge-colored graph is called a {\it proper tree} if no two adjacent edges of $T$ receive the same color. Let $G$ be a connected graph of order $n$ and $k$ be an integer with $2\leq k \leq n$. For $S\subseteq V(G)$ and $|S|…

The intersection graph of a group $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H$…

For a fixed graph $H$, the $H$-IS-Deletion problem asks, given a graph $G$, for the minimum size of a set $S \subseteq V(G)$ such that $G\setminus S$ does not contain $H$ as an induced subgraph. Motivated by previous work about hitting…

A set $D$ of vertices in a graph $G$ is a dominating set if every vertex of $G$, which is not in $D$, has a neighbor in $D$. A set of vertices $D$ in $G$ is convex (respectively, isometric), if all vertices in all shortest paths…

Let $G=(V,E)$ be a graph. An ordering of $G$ is a bijection $\alpha: V\dom \{1,2,..., |V|\}.$ For a vertex $v$ in $G$, its closed neighborhood is $N[v]=\{u\in V: uv\in E\}\cup \{v\}.$ The profile of an ordering $\alpha$ of $G$ is…

A hole in a graph is an induced subgraph which is a cycle of length at least four. A graph is chordal if it contains no holes. Following McKee and Scheinerman (1993), we define the chordality of a graph $G$ to be the minimum number of…

Let $t$ be a positive real number. A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components, and all graphs are considered 0-tough. The toughness of a graph is the largest…

For any finite set $\mathcal{H} = \{H_1,\ldots,H_p\}$ of graphs, a graph is $\mathcal{H}$-subgraph-free if it does not contain any of $H_1,\ldots,H_p$ as a subgraph. In recent work, meta-classifications have been studied: these show that if…

We introduce a new Steiner-type problem for directed graphs named \textsc{$q$-Root Steiner Tree}. Here one is given a directed graph $G=(V,A)$ and two subsets of its vertices, $R$ of size $q$ and $T$, and the task is to find a minimum size…

A graph $G$ is called a replication graph of a graph $H$ if $G$ is obtained from $H$ by replacing vertices of $H$ by arbitrary cliques of vertices and then replacing each edge in $H$ by all the edges between corresponding cligues. For a…

Recently, the authors gave Ramsey-type results for the path cover/partition number of graphs. In this paper, we continue the research about them focusing on digraphs, and find a relationship between the path cover/partition number 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…

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…

In a graph $G=(V,E)$, a module is a vertex subset $M$ of $V$ such that every vertex outside $M$ is adjacent to all or none of $M$. For example, $\emptyset$, $\{x\}$ $(x\in V )$ and $V$ are modules of $G$, called trivial modules. A graph,…

We present exact and heuristic algorithms that find, for a given family of graphs, a graph that contains each member of the family as an induced subgraph. For $0 \leq k \leq 6$, we give the minimum number of vertices $f(k)$ in a graph…

An \emph{asteroidal triple} is a set of three independent vertices in a graph such that any two vertices in the set are connected by a path which avoids the neighbourhood of the third. A classical result by Lekkerkerker and Boland \cite{6}…

Undirected co-graphs are those graphs which can be generated from the single vertex graph by disjoint union and join operations. Co-graphs are exactly the P_4-free graphs (where P_4 denotes the path on 4 vertices). Co-graphs itself and…

The age $\mathcal{A}(G)$ of a graph $G$ (undirected and without loops) is the collection of finite induced subgraphs of $G$, considered up to isomorphy and ordered by embeddability. It is well-quasi-ordered (wqo) for this order if it…

A bipartite graph $G=(A,B,E)$ is ${\cal H}$-convex, for some family of graphs ${\cal H}$, if there exists a graph $H\in {\cal H}$ with $V(H)=A$ such that the set of neighbours in $A$ of each $b\in B$ induces a connected subgraph of $H$.…