Related papers: The decycling numbers of graphs

The deck of a graph $G$ is given by the multiset of (unlabelled) subgraphs $\{G-v:v\in V(G)\}$. The subgraphs $G-v$ are referred to as the cards of $G$. Brown and Fenner recently showed that, for $n\geq29$, the number of edges of a graph…

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 {\em $(d,h)$-decomposition} of a graph $G$ is an order pair $(D,H)$ such that $H$ is a subgraph of $G$ where $H$ has the maximum degree at most $h$ and $D$ is an acyclic orientation of $G-E(H)$ of maximum out-degree at most $d$. A graph…

The independent domination number $i(G)$ of a graph $G$ is the minimum cardinality of a maximal independent set of $G$, also called an $i(G)$-set. The $i$-graph of $G$, denoted $\mathscr{I}(G)$, is the graph whose vertices correspond to the…

The metric (resp. edge metric) dimension of a simple connected graph $G$, denoted by dim$(G)$ (resp. edim$(G)$), is the cardinality of a smallest vertex subset $S\subseteq V(G)$ for which every two distinct vertices (resp. edges) in $G$…

Let $G$ be a graph. We introduce the acyclic b-chromatic number of $G$ as an analogue to the b-chromatic number of $G$. An acyclic coloring of a graph $G$ is a map $c:V(G)\rightarrow \{1,\dots,k\}$ such that $c(u)\neq c(v)$ for any $uv\in…

A detour of a graph G is a longest path in G. The detour order of G is the number of vertices in a detour of G. A graph is said to be detour-saturated if the addition of any edge increases strictly the detour order. L.W. Beineke, J.E.…

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex (edge) labeling with $d$ labels that is preserved only by the trivial automorphism. It is known that for every graph $G$…

An $H$-decomposition of $G$ is a partition of the edge-set of $G$ into subsets, where each subset induces a copy of the graph $H$. A $k$-orthogonal $H$-decomposition of a graph $G$ is a set of $k$ $H$-decompositions of $G$, such that any…

A set S of vertices of a graph G is a geodesic transversal of G if every maximal geodesic of G contains at least one vertex of S. We determine a smallest geodesic transversal in certain interconnection networks such as mesh of trees, and…

The crossing number of a graph $G$ is the least number of crossings over all possible drawings of $G$. We present a structural characterization of graphs with crossing number one.

Given a (directed) graph G=(V,A), a subset X of V is an interval of G provided that for any a, b\in X and x\in V-X, (a,x)\in A if and only if (b,x)\in A and (x,a)\in A if and only if (x,b)\in A. For example, \emptyset, \{x\} (x \in V) and V…

We introduce a graph decomposition which exists for all simple, connected graphs $G=(V,E)$. The decomposition $V = A \cup B \cup C$ is such that each vertex in $A$ has more neighbors in $B$ than in $A$ and vice versa. $C$ is `balanced':…

For a graph $G$ and $a,b\in V(G)$, the shortest path reconfiguration graph of $G$ with respect to $a$ and $b$ is denoted by $S(G,a,b)$. The vertex set of $S(G,a,b)$ is the set of all shortest paths between $a$ and $b$ in $G$. Two vertices…

A strict bramble of a graph $G$ is a collection of pairwise-intersecting connected subgraphs of $G.$ The order of a strict bramble ${\cal B}$ is the minimum size of a set of vertices intersecting all sets of ${\cal B}.$ The strict bramble…

In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V (G)[E(G) is called the mixed metric dimension of G. In this paper we first establish the exact value of the mixed metric dimension…

A cyclic base ordering of a connected graph $G$, is a cyclic ordering of $E(G)$ such that every cyclically consecutive $|V(G)|-1$ edges form a spanning tree. In this project, we study cyclic base ordering of various families of graphs,…

The fixing number of a graph $G$ is the order of the smallest subset $S$ of its vertex set $V(G)$ such that stabilizer of $S$ in $G$, $\Gamma_{S}(G)$ is trivial. Let $G_{1}$ and $G_{2}$ be disjoint copies of a graph $G$, and let…

In a graph G, cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) is the (vertex) metric dimension of G. Similarly, the cardinality of such a set is the edge metric dimension of G, if it…

Given a graph $G=(V,E)$, a subset $X$ of $V$ is an interval of $G$ provided that for any $a, b\in X$ and $ x\in V \setminus X$, $\{a,x\}\in E$ if and only if $\{b,x\}\in E$. For example, $\emptyset$, $\{x\}(x\in V)$ and $V$ are intervals of…