Related papers: Bounds on Independent Isolation in Graphs
An isolating set in a graph is a set $X$ of vertices such that every edge of the graph is incident with a vertex of $X$ or its neighborhood. The isolation number of a graph, or equivalently the vertex-edge domination number, is the minimum…
An isolating set in a graph $G$ is a set $S$ of vertices such that removing $S$ and its neighborhood leaves no edge. The isolation number $\iota(G)$ of $G$ (also known as the vertex-edge domination number) is the minimum size among all…
A set $D$ of vertices of a graph $G$ is isolating if the set of vertices not in $D$ or with no neighbor in $D$ is independent. The isolation number of $G$, denoted by $\iota (G)$, is the minimum cardinality of an isolating set of $G$. It is…
For any graph $G=(V,E)$, a subset $S\subseteq V$ is called {\it an isolating set} of $G$ if $V\setminus N_G[S]$ is an independent set of $G$, where $N_G[S]=S\cup N_G(S)$, and {\it the isolation number} of $G$, denoted by $\iota(G)$, is the…
An independent dominating set of a graph, also known as a maximal independent set, is a set $S$ of pairwise non-adjacent vertices such that every vertex not in $S$ is adjacent to some vertex in $S$. We prove that for $\Delta=4$ or…
We define an all-$k$-isolating set of a graph to be a set $S$ of vertices such that, if one removes $S$ and all its neighbors, then no component in what remains has order $k$ or more. The case $k=1$ corresponds to a dominating set and the…
Given a set $\mathcal{F}$ of graphs, we call a copy of a graph in $\mathcal{F}$ an $\mathcal{F}$-graph. The $\mathcal{F}$-isolation number of a graph $G$, denoted by $\iota(G,\mathcal{F})$, is the size of a smallest set $D$ of vertices of…
Given a set $\mathcal{F}$ of graphs, we call a copy of a graph in $\mathcal{F}$ an $\mathcal{F}$-graph. The $\mathcal{F}$-isolation number of a graph $G$, denoted by $\iota(G,\mathcal{F})$, is the size of a smallest set $D$ of vertices of…
A dissociation set in a graph is a subset of vertices which induces a subgraph with maximum degree at most one. The dissociation number of a graph is the maximum cardinality of its dissociation sets. In this paper, we consider the…
For a connected $n$-vertex graph $G$ and a set $\mathcal{F}$ of graphs, let $\iota(G,\mathcal{F})$ denote the size of a smallest set $D$ of vertices of $G$ such that the graph obtained from $G$ by deleting the closed neighbourhood of $D$…
The isolation number of a graph $G$ (also called the vertex-edge domination number of $G$), denoted by $\iota(G)$, is the size of a smallest subset $D$ of the vertex set $V(G)$ of $G$ such that $G-N[D]$ (the graph obtained by deleting the…
Given a set $\mathcal{F}$ of graphs, we call a copy of a graph in $\mathcal{F}$ an $\mathcal{F}$-graph. The $\mathcal{F}$-isolation number of a graph $G$, denoted by $\iota(G,\mathcal{F})$, is the size of a smallest subset $D$ of the vertex…
A maximal independent set in a graph $G$ is an independent set that cannot be extended to a larger independent set by adding any vertex from $G$. This paper investigates the problem of determining the maximum number of maximal independent…
Let $G$ be a graph and $k \geq 3$ an integer. A subset $D \subseteq V(G)$ is a $k$-clique (resp., cycle) isolating set of $G$ if $G-N[D]$ contains no $k$-clique (resp., cycle). In this paper, we prove that every connected graph with maximum…
Extremal problems related to the enumeration of graph substructures, such as independent sets, matchings, and induced matchings, have become a prominent area of research with the advancement of graph theory. A subset of vertices is called a…
The dissociation number ${\rm diss}(G)$ of a graph $G$ is the maximum order of a set of vertices of $G$ inducing a subgraph that is of maximum degree at most $1$. Computing the dissociation number of a given graph is algorithmically hard…
A subset of vertices in a graph $G$ is considered a maximal dissociation set if it induces a subgraph with vertex degree at most 1 and it is not contained within any other dissociation sets. In this paper, it is shown that for $n\geq 3$,…
This paper introduces the notion of an $(\iota,q)$-critical graph. The isolation number of a graph $G$, denoted by $\iota(G)$ and also known as the vertex-edge domination number of $G$, is the size of a smallest subset $D$ of the vertex set…
The 'separation dimension' of a graph $G$ is the smallest natural number $k$ for which the vertices of $G$ can be embedded in $\mathbb{R}^k$ such that any pair of disjoint edges in $G$ can be separated by a hyperplane normal to one of the…
A set $V$ is said to be separated by subsets $V_1,\ldots,V_k$ if, for every pair of distinct elements of $V$, there is a set $V_i$ that contains exactly one of them. Imposing structural constraints on the separating subsets is often…