Suzanne M. Seager

Some remarks on domination: SOME REMARKS ON DOMINATION

Proof: The proof is by induction on |V (G)| + |E(G)|. The smallest graph as described in the lemma is K1,3, for which the statement holds. This gives the start of our induction. Let x = |X| and y = |Y |. If there exists a vertex v in Y of degree at least 4, then delete any edge e incident to v. The subset A of G− e guaranteed by the inductive hypothesis is adjacent in G to every vertex in Y as desired. So we may assume that the vertices in Y are all of

The double competition number of some triangle-free graphs

Abstract The competition graph of a digraph was introduced by Joel Cohen in 1968 in the study of ecological niches. It was generalized by Debra Scott in 1985 to the competition-common enemy graph. In this paper, we study some triangle-free competition-common enemy graphs.

Locating a robber on a graph

Abstract Consider the following game of a cop locating a robber on a connected graph. At each turn, the cop chooses a vertex of the graph to probe and receives the distance from the probe to the robber. If she can uniquely locate the robber after this probe, then she wins. Otherwise the robber may either stay put or move to any vertex adjacent to his location other than the probe vertex. The cop’s goal is to minimize the number of probes required to locate the robber, while the robber’s goal is to avoid being located. This is a synthesis of the cop and robber game with the metric dimension problem. We analyse this game for several classes of graphs, including cycles and trees.

Locating a backtracking robber on a tree

Abstract Carraher, Choi, Delcourt, Erickson, and West (2012) [4] introduced the following version of a robber locating game: At each turn, the cop chooses a vertex of the graph to probe, and receives the distance from the probe to the robber. The cop wins if she can uniquely locate the robber after this probe. Otherwise the robber may stay put or move to any vertex adjacent to his location. We answer some of their conjectures and characterize the trees for which the cop wins.

The chip firing game onn-cycles

Spencer introduced a chip firing game on a line, which was generalized by Björner, Lovasz and Shor to graphs. This paper characterizes those configurations which may be repeated in a game, and analyzes the game forn-cycles.

The Domination Number of Cubic Graphs with Girth at least Five

Abstract Let γ (G) be the domination number of a graph G. We show that if G has maximum degree at most 3 and girth at least 5, then γ(G) ≤ 1 7 (4n − e + p + 3i) where G has n nodes, e edges, p pendant nodes, and i isolated nodes. It follows that if G is a cubic graph with girth at least 5, then γ (G) ≤ 5 14 n .

Analysis Boot Camp: An Alternative Path to Epsilon-Delta Proofs in Real Analysis

Abstract For many of my students, Real Analysis I is the first, and only, analysis course they will ever take, and these students tend to be overwhelmed by epsilon-delta proofs. To help them I reordered Real Analysis I to start with an “Analysis Boot Camp” in the first 2 weeks of class, which focuses on working with inequalities, absolute value, and multiple quantifiers. This helps with all the topics which follow, and when we finally reach limits of real-valued functions, the weaker students find epsilon-delta proofs much easier to handle, and are more likely to pass the final exam.

The greedy algorithm for domination in graphs of maximum degree 3

We show that for a connected graph with n nodes and e edges and maximum degree at most 3, the size of the dominating set found by the greedy algorithm is at most 10n - 2e/13 if e ≤ 11/10n, 11n - 4e/11 if 11/10n ≤ e ≤ 11/8n, and 10n - 4e/9 if e ≥ 11/8n.