Connectivity of Graphs Induced by Directional Antennas

This paper addresses the problem of finding an orientation and a minimum radius for directional antennas of a fixed angle placed at the points of a planar set S, that induce a strongly connected communication graph. We consider problem instances in which antenna angles are fixed at 90 and 180 degrees, and establish upper and lower bounds for the minimum radius necessary to guarantee strong connectivity. In the case of 90-degree angles, we establish a lower bound of 2 and an upper bound of 7. In the case of 180-degree angles, we establish a lower bound of sqrt(3) and an upper bound of 1+sqrt(3). Underlying our results is the assumption that the unit disk graph for S is connected.Comment: 8 pages, 10 figure

Rainbow Connection Number and Connected Dominating Sets

Rainbow connection number rc(G) of a connected graph G is the minimum number of colours needed to colour the edges of G, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this paper we show that for every connected graph G, with minimum degree at least 2, the rainbow connection number is upper bounded by {\gamma}_c(G) + 2, where {\gamma}_c(G) is the connected domination number of G. Bounds of the form diameter(G) \leq rc(G) \leq diameter(G) + c, 1 \leq c \leq 4, for many special graph classes follow as easy corollaries from this result. This includes interval graphs, AT-free graphs, circular arc graphs, threshold graphs, and chain graphs all with minimum degree at least 2 and connected. We also show that every bridge-less chordal graph G has rc(G) \leq 3.radius(G). In most of these cases, we also demonstrate the tightness of the bounds. An extension of this idea to two-step dominating sets is used to show that for every connected graph on n vertices with minimum degree {\delta}, the rainbow connection number is upper bounded by 3n/({\delta} + 1) + 3. This solves an open problem of Schiermeyer (2009), improving the previously best known bound of 20n/{\delta} by Krivelevich and Yuster (2010). Moreover, this bound is seen to be tight up to additive factors by a construction of Caro et al. (2008).Comment: 14 page

Aspects of distance measures in graphs.

Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2011.In this thesis we investigate bounds on distance measures, namely, Steiner diameter and radius, in terms of other graph parameters. The thesis consists of four chapters. In Chapter 1, we define the most significant terms used throughout the thesis, provide an underlying motivation for our research and give background in relevant results. Let G be a connected graph of order p and S a nonempty set of vertices of G. Then the Steiner distance d(S) of S is the minimum size of a connected subgraph of G whose vertex set contains S. If n is an integer, 2 ≤ n ≤ p, the Steiner n-diameter, diamn(G), of G is the maximum Steiner distance of any n-subset of vertices of G. In Chapter 2, we give a bound on diamn(G) for a graph G in terms of the order of G and the minimum degree of G. Our result implies a bound on the ordinary diameter by Erdös, Pach, Pollack and Tuza. We obtain improved bounds on diamn(G) for K3-free graphs and C4-free graphs. In Chapter 3, we prove that, if G is a 3-connected plane graph of order p and maximum face length l then the radius of G does not exceed p/6 + 5l/6 + 5/6. For constant l, our bound improves on a bound by Harant. Furthermore we extend these results to 4- and 5-connected planar graphs. Finally, we complete our study in Chapter 4 by providing an upper bound on diamn(G) for a maximal planar graph G

Rainbow Connection Number and Radius

The rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour its edges, so that every pair of its vertices is connected by at least one path in which no two edges are coloured the same. In this note we show that for every bridgeless graph G with radius r, rc(G) <= r(r + 2). We demonstrate that this bound is the best possible for rc(G) as a function of r, not just for bridgeless graphs, but also for graphs of any stronger connectivity. It may be noted that for a general 1-connected graph G, rc(G) can be arbitrarily larger than its radius (Star graph for instance). We further show that for every bridgeless graph G with radius r and chordality (size of a largest induced cycle) k, rc(G) <= rk. It is known that computing rc(G) is NP-Hard [Chakraborty et al., 2009]. Here, we present a (r+3)-factor approximation algorithm which runs in O(nm) time and a (d+3)-factor approximation algorithm which runs in O(dm) time to rainbow colour any connected graph G on n vertices, with m edges, diameter d and radius r.Comment: Revised preprint with an extra section on an approximation algorithm. arXiv admin note: text overlap with arXiv:1101.574