We study the Steiner Tree problem on unit disk graphs. Given a n vertex graph G, subset $$R\subseteq V(G)$$ of t vertices and positive integer k, objective is to decide if there exists tree T in G that spans over all R uses at most k from $$V\setminus R$$ . The are referred as terminals $$V(G)\setminus vertices. First, we show NP-hard. Next, prove graphs can be solved $$n^{O(\sqrt{t+k})}$$ time...

Klav́ık et al. [arXiv:1207.6960] recently introduced a generalization of recognition called the bounded representation problem which we study for the classes of interval and proper interval graphs. The input gives a graph G and in addition for each vertex v two intervals Lv and Rv called bounds. We ask whether there exists a bounded representation in which each interval Iv has its left endpoint ...

We study the problem of routing in three-dimensional ad hoc networks. We are interested in routing algorithms that guarantee delivery and are k-local, i.e., each intermediate node v’s routing decision only depends on knowledge of the labels of the source and destination nodes, of the subgraph induced by nodes within distance k of v, and of the neighbour of v from which the message was received....

the first ($pi_1$) and the second $(pi_2$) multiplicative zagreb indices of a connected graph $g$, with vertex set $v(g)$ and edge set $e(g)$, are defined as $pi_1(g) = prod_{u in v(g)} {d_u}^2$ and $pi_2(g) = prod_{uv in e(g)} {d_u}d_{v}$, respectively, where ${d_u}$ denotes the degree of the vertex $u$. in this paper we present a simple approach to order these indices for connected graphs on ...