Peter J. Slater

Centers to centroids in graphs

For S contained in V(G) the S-center and S-centroid of G are defined as the collection of vertices u an element of V(G) that minimize e/sub s/(u) = max (d(u,v): v an element of S) and d/sub s/(u) = ..sigma../sub v an element of S/d(u,v), respectively. This procedure generalizes the standard definition of center and centroid from the special case of S = V(G). For 1 less than or equal to k less than or equal to absolute value V(G) and u an element of V(G) let r/sub k/(u) = max (..sigma../sub s an element of S/d(u,s): S contained in V(G), absolute value S = k). The k-centrum of G, denoted C(G;k), is defined to be the subset of vertices u in G for which r/sub k/(u) is a minimum. This approach also generalizes the standard definitions of center and centroid since C(G;1) is the center of C(G; absolute value V(G)) is the centroid. The structure of these sets for trees is examined. Generalizations of theorems of Jordan and Zelinka are included. 4 figures, 2 tables.

R -Domination in Graphs

The problem of finding a minimum <italic>k</italic>-basis of graph <italic>G</italic> is that of selecting as small a set <italic>B</italic> of vertices as possible such that every vertex of <italic>G</italic> is at distance <italic>k</italic> or less from some vertex in <italic>B</italic>. Cockayne, Goodman, and Hedetniemi previously developed a linear algorithm to find a minimum 1-basis (a minimum dominating set) when <italic>G</italic> is a tree. In this paper the <italic>k</italic>-basis problem is placed in a more general setting, and a linear algorithm is presented that solves the problem for any forest.

On graphs containing a given graph as center

We examine the problem of embedding a graph H as the center of a supergraph G, and we consider what properties one can restrict G to have. Letting A(H) denote the smallest difference ∣V(G)∣ - ∣V(H)∣ over graphs G having center isomorphic to H it is demonstrated that A(H) ≤ 4 for all H, and for 0 ≤ i ≤ 4 we characterize the class of trees T with A(T) = i. for n ≥ 2 and any graph H, we demonstrate a graph G with point and edge connectivity equal to n, with chromatic number X(G) = n + X(H), and whose center is isomorphic to H. Finally, if ∣V(H)∣ ≥ 9 and k ≥ ∣V(H)∣ + 1, then for n sufficiently large (with n even when k is odd) we can construct a k-regular graph on n vertices whose center is isomorphic to H.

Medians of arbitrary graphs

For each vertex u in a connected graph H, the distance of u is the sum of the distances from u to each of the vertices v of H. A vertex of minimum distance in H is called a median vertex. It is shown that for any graph G there exists a graph H for which the subgraph of H induced by the median vertices is isomorphic to G.

Gossips and telegraphs

Suppose we have a group of n people, each possessing an item of information not known to any of the others and that during each unit of time each person can send all of the information he knows to at most other people. Further suppose that each of at most k other people can send all of the information they know to him. Determine the length of time required, f(n, k), so that all n people know all of the information. We show f(n, k) = ⌜logk+1n⌝. n nWe define g(n, k) analogously except that no person may both send and receive information during a unit time period. We show ⌜logk+1n⌝≤g(n, k)≤2⌜logk+1n⌝ in general and further show that the upper bound can be significantly improvea in the cases k = 1 or 2. We conjecture g(n, k) = bk logk+1n+0(1) for a function bk we determine.

Geodetic connectivity of graphs

A graph G is said to be n -geodetically connected if and only if G is connected and the removal of at least n points is required to increase the distance between any pair of points. Geodetic analogs of results such as Mengers theorem and Diracs fan theorem are shown to hold. Some other characterizations of n -geodetically connected graphs are obtained, one of which shows geodetic connectivity to be a local property in contrast to the usual connectivity.

On k-critical, n-connected graphs☆

A graph G which is n-connected (but not (n + 1)-connected)is defined to be k-critical if for every S ⊆ V(G), where |S|⩽k, the connectivity of G − S is n − |S|. We will say that G is an (n∗,k∗) graph if G is n-connected (but not (n + 1)-connected) and k-critical (but not (k + 1)- critical). This initial study of k-critical graphs is concerned with the problem of determining the values of n and k for which there exists an (n∗, k∗) graph.

A classification of 4-connected graphs

Having observed Tuttes classification of 3-connected graphs as those attainable from wheels by line addition and point splitting and Hedetniemis classification of 2-connected graphs as those obtainable from K2 by line addition, subdivision and point addition, one hopes to find operations which classify n-connected graphs as those obtainable from, for example, Kn+1. In this paper I give several generalizations of the above operations and use Halins theorem to obtain two variations of Tuttes theorem as well as a classification of 4-connected graphs.

Counterexamples to randić's conjecture on distance degree sequences for trees

As counterexamples to a conjecture of Randic, pairs of nonisomorphic trees with the same collections of distance degree sequences are presented.

Matchings and transversals in hypergraphs, domination and independence-in trees

Abstract A family of hypergraphs is exhibited which have the property that the minimum cardinality of a transversal is equal to the maximum cardinality of a matching. A result concerning domination and independence in trees which generalises a recent result of Meir and Moon is deduced.