15322039
Tree coloring of distance graphs with a real interval set
Let RR be the set of real numbers and DD be a subset of the positive real numbers. The distance graph  G(R,D)G(R,D) is a graph with the vertex set RR and two vertices xx and yy are adjacent if and only if |x−y|∈D|x−y|∈D. In this work, the vertex arboricity (i.e., the minimum number of subsets into which the vertex set V(G)V(G) can be partitioned so that each subset induces an acyclic subgraph) of G(R,D)G(R,D) is determined for DD being an interval between 1 and δδ.
45978330
Two sufficient conditions for a graph to be type 1
32836256
Minimum average distance of strong orientations of graphs
The average distance of a graph (strong digraph) G, denoted by µ(G) is the average, among the distances between all pairs (ordered pairs) of vertices of G. If G is a 2-edge-connected graph, then µ→min(G) is the minimum average distance taken over all strong orientations of G. A lower bound for µ→min(G) in terms of the order, size, girth and average distance of G is established and shown to be sharp for several complete multipartite graphs. It is shown that there is no upper bound for µ→min(G) in terms of µ(G). However, if every edge of G lies on 3-cycle, then it is shown that µ→min(G) ≤ 7/4 µ(G). This bound is improved for maximal planar graphs to 5/3 µ(G) and even further to 3/2 µ(G) for eulerian maximal planar graphs and for outerplanar graphs with the property that every edge lies on 3-cycle. In the last case the bound is shown to be sharp.
2394010
Efficient broadcasting in ad hoc wireless networks using directional antennas
Using directional antennas to conserve bandwidth and energy consumption in ad hoc wireless networks (or simply ad hoc networks) is becoming popular. However, applications of directional antennas for broadcasting have been limited. We propose a novel broadcast protocol called directional self-pruning (DSP) for ad hoc wireless networks using directional antennas. DSP is a nontrivial generalization of an existing localized deterministic broadcast protocol using omnidirectional antennas. Compared with its omnidirectional predecessor, DSP uses about the same number of forward nodes to relay the broadcast packet, while the number of forward directions that each forward node uses in transmission is significantly reduced. With the lower broadcast redundancy, DSP is more bandwidth and energy-efficient. DSP is based on 2-hop neighborhood information and does not rely on location or angle-of-arrival (AoA) information. Two special cases of DSP are discussed: the first one preserves shortest paths in reactive routing discoveries; the second one uses the directional reception mode to minimize broadcast redundancy. DSP is a localized protocol. Its expected number of forward nodes is O(1) times the optimal value. An extensive simulation study using both custom and ns2 simulators show that DSP significantly outperforms both omnidirectional broadcast protocols and existing directional broadcast protocols.
204989501
On clique partitions of split graphs
Abstract Split graphs are graphs formed by taking a complete graph and an empty graph disjoint from it and some or all of the possible edges joining the two. We prove that the problem of deciding the clique partition number is NP-complete, even when restricted to the class of split graphs.
31006771
Extended Dominating Set in Ad Hoc Networks Using Cooperative Communication
We propose a notion of extended dominating set whereby each node in an ad hoc network is covered by either a dominating neighbor or several 2-hop dominating neighbors. This work is motivated by cooperative communication in ad hoc networks where transmitting independent copies of a packet generates diversity and combats the effects of fading. In this paper we propose several efficient heuristic algorithms for constructing a small extended dominating set.
6186942
The number of removable edges in a 4-connected graph
Let G be a 4-connected graph. For an edge e of G, we do the following operations on G: first, delete the edge e from G, resulting the graph G - e; second, for all the vertices x of degree 3 in G - e, delete x from G - e and then completely connect the 3 neighbors of x by a triangle. If multiple edges occur, we use single edges to replace them. The final resultant graph is denoted by G ⊖ e. If G ⊖ e is still 4-connected, then e is called a removable edge of G. In this paper we prove that every 4-connected graph of order at least six (excluding the 2-cyclic graph of order six) has at least (4|G|+16)/7 removable edges. We also give the structural characterization of 4-connected graphs for which the lower bound is sharp.
10245995
Resource allocation in wireless networks using directional antennas
With the continued increase of speed and capacities of computing devices and the growing needs of people for mobile computing capabilities, Mobile ad hoc networks (MANETs) have gained a lot of interest from the research community. Quality of service (QoS) provisioning in MANETs is an essential component that is needed to support multimedia and real-time applications. On the other hand, directional antenna technology provides the capability for considerable increase in spatial reuse which is essential in the wireless medium. In this paper, a bandwidth reservation protocol for QoS routing in TDMA-based MANETs using directional antennas is presented. The routing algorithm allows a source node to reserve a path to a particular destination with the needed bandwidth which is represented by the number of slots in the data phase of the TDMA frame. Further optimizations to improve the efficiency and resource utilization of the network are provided