Karen Seyffarth

Domination numbers of planar graphs

The problem of determining the domination number of a graph is a well known NP-hard problem, even when restricted to planar graphs. By adding a further restriction on the diameter of the graph, we prove that planar graphs with diameter two and three have bounded domination numbers. This implies that the domination number of such a graph can be determined in polynomial time. We also give examples of planar graphs of diameter four, and nonplanar graphs of diameter two, having arbitrarily large domination numbers.

Dynamic and self-stabilizing distributed matching

Finding a maximal or maximum matching in a graph is a well-understood problem for which efficient sequential algorithms exist. Applications of matchings in distributed settings make it desirable to find self-stabilizing asynchronous distributed solutions to these problems. We first present a self-stabilizing algorithm for finding a maximal matching in a general anonymous network under read/write atomicity with linear round complexity. This is followed by a self-stabilizing algorithm, with quadratic time complexity, for finding a maximum matching in a bipartite network under composite atomicity. These results represent significant progress in the area of distributed algorithms for matchings.

Large planar graphs with given diameter and maximum degree

Abstract We consider the problem of determining the maximum number of vertices in a planar graph with given maximum degree Δ and diameter k. This number has previously been exactly determined when k = 2. We show here that when k = 3, the number is roughly between 4.5Δ and 8Δ. We also show that in general the number is Θ(Δ ⌊ k 2 ⌋ ) for any fixed value of k.

The k-Dominating Graph

Given a graph G, the k-dominating graph of G, Dk(G), is defined to be the graph whose vertices correspond to the dominating sets of G that have cardinality at most k. Two vertices in Dk(G) are adjacent if and only if the corresponding dominating sets of G differ by either adding or deleting a single vertex. The graph Dk(G) aids in studying the reconfiguration problem for dominating sets. In particular, one dominating set can be reconfigured to another by a sequence of single vertex additions and deletions, such that the intermediate set of vertices at each step is a dominating set if and only if they are in the same connected component of Dk(G). In this paper we give conditions that ensure Dk(G) is connected.

Largest planar graphs of diameter two and fixed maximum degree

We compute the exact maximum number of vertices in a planar graph with diameter two and maximum degree ?, for any ??8. Results for larger diameter are also mentioned.

Maximal planar graphs of diameter two

A maximal planar graph is a simple planar graph in which every face is a triangle. We show here that such graphs with maximum degree Δ and diameter two have no more than 3/2Δ + 1 vertices. We also show that there exist maximal planar graphs with diameter two and exactly [3/2Δ + 1] vertices.

Constructions of large planar networks with given degree and diameter

There is considerable interest in constructing large networks with given diameter and maximum degree. In certain applications, there is a natural restriction for the networks to be planar. Thus, consider the problem of determining the maximum number of nodes in a planar network with maximum degree Δ and diameter at most k. We have previously proved that this number is at most (roughly) 12kΔ [k/2] and there is a trivial lower bound of about (Δ - 1) [k/2] . We introduce a number of general constructions which substantially improve the lower bound and yield the largest known networks. We also provide a catalog of the best-known networks for small values of Δ and k, many obtained by specialized constructions.

Hajo´s' conjecture and small cycle double covers of planar graphs

Abstract We prove that every simple even planar graph on n vertices has a partition of its edge set into at most ⌊( n - 1)/2⌋ cycles. A previous proof of this result was given by Tao, but is incomplete, and we provide here a somewhat different proof. We also discuss the connection between this result and the Small Cycle Double Cover Conjecture .

Small cycle double covers of 4-connected planar graphs

Acycle double cover of a graph,G, is a collection of cycles,C, such that every edge ofG lies in precisely two cycles ofC. TheSmall Cycle Double Cover Conjecture, proposed by J. A. Bondy, asserts that every simple bridgeless graph onn vertices has a cycle double cover with at mostn−1 cycles, and is a strengthening of the well-knownCycle Double Cover Conjecture. In this paper, we prove Bondys conjecture for 4-connected planar graphs.

Reconfiguring k-colourings of Complete Bipartite Graphs

Let H be a graph, and k ≥ χ(H) an integer. We say that H has a cyclic Gray code of k-colourings if and only if it is possible to list all its k-colourings in such a way that consecutive colourings, including the last and the first, agree on all vertices of H except one. The Gray code number of H is the least integer k0(H) such that H has a cyclic Gray code of its k-colourings for all k ≥ k0(H). For complete bipartite graphs, we prove that k0(K`,r) = 3 when both ` and r are odd, and k0(K`,r) = 4 otherwise.