Related papers: Breaking Symmetry in Graphs by Resolving Sets

An edge-colored graph $G$, where adjacent edges may have the same color, is {\it rainbow connected} if every two vertices of $G$ are connected by a path whose edge has distinct colors. A graph $G$ is {\it $k$-rainbow connected} if one can…

An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. The smallest size of an identifying code of $G$ is denoted $\gamma^{\text{ID}}(G)$.…

A path in an edge-colored graph, where adjacent edges may be colored the same, is a rainbow path if no two edges of it are colored the same. A nontrivial connected graph $G$ is rainbow connected if there is a rainbow path connecting any two…

In this paper we study the following problem. Let $A$ be a fixed graph, and let $\hom(G,A)$ denote the number of homomorphisms from a graph $G$ to $A$. Furthermore, let $v(G)$ denote the number of vertices of $G$, and let $\mathcal{G}_d$…

A digraph $D$ is an oriented graph if $D$ does not have a pair of opposite arcs. The degree of a vertex $v$ of $D$ is the sum of the in-degree and out-degree of $v.$ Let $fvs(D)$ be the minimum number of vertices whose deletion from $D$…

A mixed graph $\widetilde{G}$ is obtained by orienting some edges of a graph $G$, where $G$ is the underlying graph of $\widetilde{G}$. Let $r(\widetilde{G})$ be the $H$-rank of $\widetilde{G}$. Denote by $r(G)$, $\kappa(G)$, $m(G)$ and…

An $(n,m)$-graph is characterised by having $n$ types of arcs and $m$ types of edges. A homomorphism of an $(n,m)$-graph $G$ to an $(n,m)$-graph $H$, is a vertex mapping that preserves adjacency, direction, and type. The $(n,m)$-chromatic…

We introduce a broad class of equations that are described by a graph, which includes many well-studied systems. For these, we show that the number of solutions (or the dimension of the solution set) can be bounded by studying certain…

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 three subgraphs of a connected graph induced by the center, annulus and periphery are called its metric subgraphs. The main results are as follows. (1) There exists a graph of order $n$ whose metric subgraphs are all paths if and only…

A graph $G$ is divisible by a graph $H$ if the characteristic polynomial of $G$ is divisible by that of $H$. In this paper, a necessary and sufficient condition for recursive graphs to be divisible by a path is used to show that the H-shape…

A set of vertices $W$ resolves a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $W$. A metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. A bipartite graph G(n,n) is…

The distinguishing index $D'(G)$ of a graph $G$ is the least number of colors necessary to obtain an edge coloring of $G$ that is preserved only by the trivial automorphism. We show that if $G$ is a connected $\alpha$-regular graph for some…

A set $W\subseteq V(G)$ is called a resolving set, if for each two distinct vertices $u,v\in V(G)$ there exists $w\in W$ such that $d(u,w)\neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. A resolving set for $G$…

Let $G$ be a graph of order $n$. A classical upper bound for the domination number of a graph $G$ having no isolated vertices is $\lfloor\frac{n}{2}\rfloor$. However, for several families of graphs, we have $\gamma(G) \le…

For a graph $G$, the tree graph ${\cal T}_{G,t}$ has all tree subgraphs of $G$ with $t$ vertices as vertex set and two tree subgraphs are neighbors if they are edge-disjoint. Also, the $r^{th}$ cut number of $G$ is the minimum number of…

The Moore bound constitutes both an upper bound on the order of a graph of maximum degree $d$ and diameter $D=k$ and a lower bound on the order of a graph of minimum degree $d$ and odd girth $g=2k+1$. Graphs missing or exceeding the Moore…

A geometric graph is a graph drawn in the plane so that its vertices and edges are represented by points in general position and straight line segments, respectively. A vertex of a geometric graph is called pointed if it lies outside of the…

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 $n\geq 3$,…

An $n$-tuple $D=(d(1),\dots,d(n))$ is a \emph{feasible degree sequence} if there is a graph on $\{1,\dots,n\}$ such that $i$ has degree $d(i)$. Any such graph will have $m=\sum_{i=1}^n d(i)/2$ edges. Letting $G(D)$ be a graph chosen…