Metric Dimension of Amalgamation of Graphs

A set of vertices $S$ resolves a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. Let $\{G_1, G_2, \ldots, G_n\}$ be a finite collection of graphs and each $G_i$ has a fixed vertex $v_{0_i}$ or a fixed edge $e_{0_i}$ called a terminal vertex or edge, respectively. The \emph{vertex-amalgamation} of $G_1, G_2, \ldots, G_n$, denoted by $Vertex-Amal\{G_i;v_{0_i}\}$, is formed by taking all the $G_i$'s and identifying their terminal vertices. Similarly, the \emph{edge-amalgamation} of $G_1, G_2, \ldots, G_n$, denoted by $Edge-Amal\{G_i;e_{0_i}\}$, is formed by taking all the $G_i$'s and identifying their terminal edges. Here we study the metric dimensions of vertex-amalgamation and edge-amalgamation for finite collection of arbitrary graphs. We give lower and upper bounds for the dimensions, show that the bounds are tight, and construct infinitely many graphs for each possible value between the bounds.Comment: 9 pages, 2 figures, Seventh Czech-Slovak International Symposium on Graph Theory, Combinatorics, Algorithms and Applications (CSGT2013), revised version 21 December 201

On Size Bipartite and Tripartite Ramsey Numbers for The Star Forest and Path on 3 Vertices

For simple graphs G and H the size multipartite Ramsey number mj(G,H) is the smallest natural number t such that any arbitrary red-blue coloring on the edges of Kjxt contains a red G or a blue H as a subgraph. We studied the size tripartite Ramsey numbers m3(G,H) where G=mK1,n and H=P3. In this paper, we generalize this result. We determine m3(G,H) where G is a star forest, namely a disjoint union of heterogeneous stars, and H=P3. Moreover, we also determine m2(G,H) for this pair of graphs G and H

Partition dimension of disjoint union of complete bipartite graphs

For any (not necessary connected) graph $G(V,E)$ and $A\subseteq V(G)$, the distance of a vertex $x\in V(G)$ and $A$ is $d(x,A)=\min\{d(x,a): a\in A\}$. A subset of vertices $A$ resolves two vertices $x,y \in V(G)$ if $d(x,A)\neq d(y,A)$. For an ordered partition $\Lambda=\{A_1, A_2,\ldots, A_k\}$ of $V(G)$, if all $d(x,A_i)\infty$ for all $x\in V(G)$, then the representation of $x$ under $\Lambda$ is $r(x|\Lambda)=(d(x,A_1), d(x,A_2), \ldots, d(x,A_k))$. Such a partition $\Lambda$ is a resolving partition of $G$ if every two distinct vertices $x,y\in V(G)$ are resolved by $A_i$ for some $i\in [1,k]$. The smallest cardinality of a resolving partition $\Lambda$ is called a partition dimension of $G$ and denoted by $pd(G)$ or $pdd(G)$ for connected or disconnected $G$, respectively. If $G$ have no resolving partition, then $pdd(G)=\infty$. In this paper, we studied the partition dimension of disjoint union of complete bipartite graph, namely $tK_{m,n}$ where $t\geq 1$ and $m\geq n\geq 2$. We gave the necessary condition such that the partition dimension of $tK_{m,n}$ are finite. Furthermore, we also derived the necessary and sufficient conditions such that $pdd(tK_{m,n})$ is either equal to $m$ or $m+1$

On The Partition Dimension of Disconnected Graphs

For a graph G=(V,E), a partition Ω={O1,O2,"¦,Ok} of the vertex set V is called a resolving partition if every pair of vertices u,v ∈ V(G) have distinct representations under Ω. The partition dimension of G is the minimum integer k such that G has a resolving k-partition. Many results in determining the partition dimension of graphs have been obtained. However, the known results are limited to connected graphs. In this study, the notion of the partition dimension of a graph is extended so that it can be applied to disconnected graphs as well. Some lower and upper bounds for the partition dimension of a disconnected graph are determined (if they are finite). In this paper, also the partition dimensions for some classes of disconnected graphs are given

The Non-Isolated Resolving Number of Some Corona Graphs

An ordered set W = {w1, w2, ..., wk} ⊆ V(G) and a vertex v in a connected graph G, the representation of v with respect to W is the ordered k-tuple r(v|W) = (d(v, w1), d(v, w2), ..., d(v, wk)), where d(x, y) represents the distance between the vertices x and y in G. The set W is called a resolving set for G if every vertex of G has distinct representations. A resolving set with the minimum number of vertices is called a basis for G and its cardinality is called the metric dimension of G, denoted by dim(G). A resolving set W is called a non-isolated resolving set if the induced subgraph 〈W〉 has no isolated vertices. The minimum cardinality of a non-isolated resolving set of G is called the non-isolated resolving number of G, denoted by nr(G). The corona product between a graph G and a graph H, denoted by G⊙H, is a graph obtained from one copy of G and |V(G)| copies H1, H2, ..., Hn of H such that all vertices in Hi are adjacent to the i-th vertex of G. We study the non-isolated resolving sets of some corona graphs. We determine nr(G⊙H) where G is any connected graph and H is a complete graph, a cycle, or a path

Non-Isolated Resolving Sets of Corona Graphs with Some Regular Graphs

Let G be a connected, simple, and finite graph. For an ordered set W={w1,w2,…,wk}⊆V(G) and a vertex v of G, the representation of v with respect to W is the k-vector r(v|W)=(dG(v,w1),…,dG(v,wk)) . The set W is called a resolving set of G, if every two vertices of G has a different representation. A resolving set containing a minimum number of vertices is called a basis of H. The number of elements in a basis of G is called the metric dimension of G and denoted by dim(G) . In this paper, we considered a resolving set W of G where the induced subgraph of G by W does not contain an isolated vertex. Such a resolving set is called a non-isolated resolving set. A non-isolated resolving set of G with minimum cardinality is called an nr -set of G. The cardinality of an nr -set of G is called the non-isolated resolving number of G, denoted by nr(G) . Let H be a graph. The corona product graph of G with H, denoted by G⊙H , is a graph obtained by taking one copy of G and |V(G)| copies of H, namely H1,H2,…,H|V(G)| , such that the i-th vertex of G is adjacent to every vertex of Hi . If the degree of every vertex of H is k, then H is called a k-regular graph. In this paper, we determined nr(G⊙H) where G is an arbitrary connected graph of order n at least two and H is a k-regular graph of order t with k∈{t−2,t−3}

The local metric dimension of split and unicyclic graphs

A set W is called a local resolving set of G if the distance of u and v to some elements of W are distinct for every two adjacent vertices u and v in G.  The local metric dimension of G is the minimum cardinality of a local resolving set of G.  A connected graph G is called a split graph if V(G) can be partitioned into two subsets V1 and V2 where an induced subgraph of G by V1 and V2 is a complete graph and an independent set, respectively.  We also consider a graph, namely the unicyclic graph which is a connected graph containing exactly one cycle.  In this paper, we provide a general sharp bounds of local metric dimension of split graph.  We also determine an exact value of local metric dimension of any unicyclic graphs

Non-Isolated Resolving Sets of Corona Graphs with Some Regular Graphs

Let G be a connected, simple, and finite graph. For an ordered set W={w1,w2,…,wk}⊆V(G) and a vertex v of G, the representation of v with respect to W is the k-vector r(v|W)=(dG(v,w1),…,dG(v,wk)). The set W is called a resolving set of G, if every two vertices of G has a different representation. A resolving set containing a minimum number of vertices is called a basis of H. The number of elements in a basis of G is called the metric dimension of G and denoted by dim(G). In this paper, we considered a resolving set W of G where the induced subgraph of G by W does not contain an isolated vertex. Such a resolving set is called a non-isolated resolving set. A non-isolated resolving set of G with minimum cardinality is called an nr-set of G. The cardinality of an nr-set of G is called the non-isolated resolving number of G, denoted by nr(G). Let H be a graph. The corona product graph of G with H, denoted by G⊙H, is a graph obtained by taking one copy of G and |V(G)| copies of H, namely H1,H2,…,H|V(G)|, such that the i-th vertex of G is adjacent to every vertex of Hi. If the degree of every vertex of H is k, then H is called a k-regular graph. In this paper, we determined nr(G⊙H) where G is an arbitrary connected graph of order n at least two and H is a k-regular graph of order t with k∈{t−2,t−3}

Non-Isolated Resolving Sets of Corona Graphs with Some Regular Graphs

Let G be a connected, simple, and finite graph. For an ordered set W={w1,w2,…,wk}⊆V(G) and a vertex v of G, the representation of v with respect to W is the k-vector r(v|W)=(dG(v,w1),…,dG(v,wk)). The set W is called a resolving set of G, if every two vertices of G has a different representation. A resolving set containing a minimum number of vertices is called a basis of H. The number of elements in a basis of G is called the metric dimension of G and denoted by dim(G). In this paper, we considered a resolving set W of G where the induced subgraph of G by W does not contain an isolated vertex. Such a resolving set is called a non-isolated resolving set. A non-isolated resolving set of G with minimum cardinality is called an nr-set of G. The cardinality of an nr-set of G is called the non-isolated resolving number of G, denoted by nr(G). Let H be a graph. The corona product graph of G with H, denoted by G⊙H, is a graph obtained by taking one copy of G and |V(G)| copies of H, namely H1,H2,…,H|V(G)|, such that the i-th vertex of G is adjacent to every vertex of Hi. If the degree of every vertex of H is k, then H is called a k-regular graph. In this paper, we determined nr(G⊙H) where G is an arbitrary connected graph of order n at least two and H is a k-regular graph of order t with k∈{t−2,t−3}