Phyllis Zweig Chinn

Vertex domination‐critical graphs

A dominating set in a graph G is a set of vertices D such that every vertex of G is either in D or is adjacent some vertex of D. The domination number Γ(G) of G is the minimum cardinality of any dominating set. A graph is vertex domination-critical if the removal of any vertex decreases its domination number. This paper gives examples and properties of vertex domination-critical graphs, presents a method of constructing them, and poses some open questions. In the process several results for arbitrary graphs are presented.

Discovery-Method Teaching in Graph Theory

Abstract A variety of recent reports have recommended sweeping changes in content and method of teaching mathematics. While many of these reports have been geared to precollege mathematics, their implementation will require changes in the way teachers are prepared in college as well as adaptation to new abilities and attitudes that should be evident in students entering colleges in the future. Graph Theory is an area of mathematics that is particularly well-suited to non-traditional methods of teaching. It is also one of the topics recommended for inclusion in high school mathematics courses. This presentation will include information from two reports from the National Council of Teachers of Mathematics and the Mathematical Sciences Education Board, along with ideas for incorporating discovery-method learning into Graph Theory classes.

Sunlike Primal Graphs

A graph G is said to be decomposed into subgraphs H , , H2, . . . , H, if every edge of G lies in exactly one H,. In this case, G = H l + H2 + + H, is called a decomposition of G with terms H 1 , H 2 , . . . , Hn. The decomposition is trivial if a t most one of the Hi contains any edges. The graphs considered in this paper are all finite, without loops or multiple edges. We do adopt two conventions. Since we are only concerned with covering edges, to avoid certain trivialities we assume each graph has no isolated vertices and at least one edge. The other convention concerns decompositions. All decompositions considered in this work will have nonisomorphic terms. Let Q be any set of graphs and let I’ C Q. Then r is primal relative to R if

Primal graphs with small degrees

Abstract It has previously been shown that there is a unique set Π of primal graphs such that every graph has an edge-decomposition into non-isomorphic elements of Π and that the only decomposition of an element of Π into non-isomorphic elements of Π is the obvious one. Here it is shown that there are infinitely many elements of Π even among graphs having a relatively simple structure. On the other hand, within this same class of graphs, we show that ‘most’ of them are not primal.