For an ordered set { } k w w w W , , , 2 1  = of vertices and a vertex v in a connected graph G, the (metric) representation of v with respect to W is the k-vector r(v/W) = (d(v, w1), d(v, w2), ..., d(v, wk)), where d(x, y)representsthe distance between the vertices x and y. A resolving set of minimum cardinality is called a minimum resolving set or abasis and the cardinality of a basis for G is its dimension dimG. For the graph G1 = (V1, E1) and G2 = (V2, E2) their joinis denoted by G1 +G2 is the graph whose vertex set is V1∪V2 and the edge set is { } V V E E v u uv E 2 1 2 1 , : ∈ ∈ =   . In this paper, we determine the metric dimension of join of paths, paths and cycles, path and stars, complete graphs,complete graphs and paths. Keywords—Join, Metric dimension, Metric basis, Resolving set. ————————————————————

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Distance in graphs

On the Metric Dimension of Joins of Two Graphs

Pairs of a tree and a nontree graph with the same status sequence

For a connected graph G = (V, E) of order at least two, a chord of a path P is an edge joining two non-adjacent vertices of P. A path P is called a monophonic path if it is a chordless path. A set S of vertices of G is a monophonic set of G if each vertex v of G lies on an x − y monophonic path for some elements x and y in S. The minimum cardinality of a monophonic set of G is defined as the monophonic number of G, denoted by m(G). A connected monophonic set of G is a monophonic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected monophonic set of G is the connected monophonic number of G and is denoted by mc(G). We determine bounds for it and characterize graphs which realize these bounds. For any two vertices u and v in G, the monophonic distance dm(u, v) from u to v is defined as the length of a longest u − v monophonic path in G. The monophonic eccentricity em(v) of a vertex v in G is the maximum monophonic distance from v to a vertex of G. The monophonic radius radmG of G is the minimum monophonic eccentricity among the vertices of G, while the monophonic diameter diammG of G is the maximum monophonic eccentricity among the vertices of G. It is shown that for positive integers r, d and n ≥ 5 with r <  d, there exists a connected graph G with radmG =  r, diammG =  d and mc(G) =  n. Also, if a,b and p are positive integers such that 2 ≤  a <  b ≤  p, then there exists a connected graph G of order p, m(G) =  a and mc(G) =  b.

The Connected Monophonic Number of a Graph

For two vertices u and v in a connected graph G, the set I(u, v) consists of all those vertices lying on a u − v geodesic in G. For a set S of vertices of G, the union of all sets I(u, v) for u, v ∈ S is denoted by I(S). A set S is convex if I(S) = S. The convexity number con(G) is the maximum cardinality of a proper convex set in G. A convex set S is maximum if |S| = con(G). The cardinality of a maximum convex set in a graph G is the convexity number of G. For a nontrivial connected graph H , a connected graph G is an H-convex graph if G contains a maximum convex set S whose induced subgraph is 〈S〉 = H . It is shown that for every positive integer k, there exist k pairwise nonisomorphic graphs H1, H2, . . . , Hk of the same order and a graph G that is Hi-convex for all i (1 i k). Also, for every connected graph H of order k 3 with convexity number 2, it is shown that there exists an H-convex graph of order n for all n k+1. More generally, it is shown that for every nontrivial connected graph H , there exists a positive integer N and an H-convex graph of order n for every integer n N .

methods

$H$-convex graphs

The Steiner $k$-diameter $sdiam_k(G)$ of a graph $G$, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical diameter. When $k=2$, $sdiam_2(G)=diam(G)$ is the classical diameter. The problem of determining the minimum size of a graph of order $n$ whose diameter is at most $d$ and whose maximum is $\ell$ was first introduced by Erdos and Renyi. In this paper, we generalize the above problem for Steiner $k$-diameter, and study the problem of determining the minimum size of a graph of order $n$ whose Steiner $3$-diameter is at most $d$ and whose maximum is at most $\ell$.

Distance in graphs

Citations

On the Metric Dimension of Joins of Two Graphs

Pairs of a tree and a nontree graph with the same status sequence

The Connected Monophonic Number of a Graph

$H$-convex graphs

Steiner diameter, maximum degree and size of a graph

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