Wayne Goddard

Defective coloring revisited

A graph is (k, d)-colorable if one can color the vertices with k colors such that no vertex is adjacent to more than d vertices of its same color. In this paper we investigate the existence of such colorings in surfaces and the complexity of coloring problems. It is shown that a toroidal graph is (3, 2)- and (5, 1)-colorable, and that a graph of genus γ is (χγ/(d + 1) + 4, d)-colorable, where χγ is the maximum chromatic number of a graph embeddable on the surface of genus γ. It is shown that the (2, k)-coloring, for k ≥ 1, and the (3, 1)-coloring problems are NP-complete even for planar graphs. In general graphs (k, d)-coloring is NP-complete for k ≥ 3, d ≥ 0. The tightness is considered. Also, generalizations to defects of several algorithms for approximate (proper) coloring are presented.

Offensive alliances in graphs

A set S is an offensive alliance if for every vertex v in its boundary N(S)−S it holds that the majority of vertices in v’s closed neighbourhood are in S. The offensive alliance number is the minimum cardinality of an offensive alliance. In this paper we explore the bounds on the offensive alliance and the strong offensive alliance numbers (where a strict majority is required). In particular, we show that the offensive alliance number is at most 2/3 the order and the strong offensive alliance number is at most 5/6 the order.

The diameter of total domination vertex critical graphs

Abstract A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G - v is less than the total domination number of G . These graphs we call γ t -critical. If such a graph G has total domination number k , we call it k - γ t -critical. We characterize the connected graphs with minimum degree one that are γ t -critical and we obtain sharp bounds on their maximum diameter. We calculate the maximum diameter of a k - γ t -critical graph for k ⩽ 8 and provide an example which shows that the maximum diameter is in general at least 5 k / 3 - O ( 1 ) .

A survey of integrity

Abstract A communication network can be considered to be highly vulnerable to disruption if the destruction of a few elements can result in no members being able to communicate with very many others. This idea suggests the concept of the integrity of a graph—the minimum sum of the orders of a set of vertices being removed and a largest remaining component. This survey includes results on the integrity of specific families of graphs and combinations of graphs, relationships with other parameters, bounds, computational complexity, and some variations on the concept.

Weakly pancyclic graphs

The problem was posed of determining the biclique partition number of the complement of a Hamiltonian path (Monson, Rees, and Pullman, Bull. Inst. Combinatorics and Appl. 14 (1995), 17–86). We define the complement of a path P, denoted

The agreement metric for labeled binary trees

\overline{P}

Self-stabilizing protocols for maximal matching and maximal independent sets for ad hoc networks

, as the complement of P in Km,n where P is a subgraph of Km,n for some m and n. We give an exact formula for the biclique partition number of the complement of a path. In particular, we solve the problem posed in [9]. We also summarize our more general results on biclique partitions of the complement of forests.

Acyclic colorings of planar graphs

Let S be a set of n objects. A binary tree of S is a binary tree whose leaves are labeled without repetition from S. The operation of pruning a tree T is that of removing some leaves from T and suppressing all inner vertices of degree 2 which are formed by this deletion. Given two trees T and U, an agreement tree is a tree that can be obtained from T as well as from U by pruning the fewest number of leaves from the two trees. A quadratic algorithm is presented for doing this and two metrics are defined based on agreement trees.

A Synchronous Self-stabilizing Minimal Domination Protocol in an Arbitrary Network Graph

We propose two distributed algorithms to maintain, respectively, a maximal matching and a maximal independent set in a given ad hoc network; our algorithms are fault tolerant (reliable) in the sense that the algorithms can detect occasional link failures and/or new link creations in the network (due to mobility of the hosts) and can readjust the global predicates. We provide time complexity analysis of the algorithms in terms of the number of rounds needed for the algorithm to stabilize after a topology change, where a round is defined as a period of time in which each node in the system receives beacon messages from all its neighbors. In any ad hoc network, the participating nodes periodically transmit beacon messages for message transmission as well as to maintain the knowledge of the local topology at the node; as a result, the nodes get the information about their neighbor nodes synchronously (at specific time intervals). Thus, the paradigm to analyze the complexity of the self-stabilizing algorithms in the context of ad hoc networks is very different from the traditional concept of an adversary daemon used in proving the convergence and correctness of self-stabilizing distributed algorithms in general.

Crossing families

Abstract It is shown that a planar graph can be partitioned into three linear forests. The sharpness of the result is also considered.