Edward T. Ordman

Clique partitions and clique coverings

Several new tools are presented for determining the number of cliques needed to (edge-)partition a graph. For a graph on n vertices, the clique partition number can grow end times as fast as the clique covering number, where c is at least 1/64. If in a clique on n vertices, the edges between cn a vertices are deleted,1/2⩽a≤1, then the number of cliques needed to partition what is left is asymptotic to c 2 n 2a ; this fills in a gap between results of Wallis for a≤1/2 and Pullman and Donald for a=1, c≥1/2. Clique coverings of a clique minus a matching are also investigated.

Minimal threshold separators and memory requirements for synchronization

Suppose that in a system of asynchronous parallel processes, certain pairs of processes mutually exclude one another (must not be in their critical sections simultaneously). This situation is modeled by a graph in which each process is represented by a vertex and each mutually excluding pair is represented by an edge. Henderson and Zalcstein have observed that if this graph is a threshold graph, then mutual exclusion can be managed by simple entrance and exit protocols using

Bounds on threshold dimension and disjoint threshold coverings

{\bf PV}

The size of chordal, interval and threshold subgraphs

-chunk operations on a single shared variable whose possible values range from zero to t, the minimal threshold separator number of the graph. A new expression is given for this separator t of a threshold graph in terms of the normal decomposition of the threshold graph given by Zalcstein and Henderson. It is shown that

Limitations of using tokens for mutual exclusion

t + 1

Generalized mutual exclusion with semaphores only

values would be needed in the shared variable even if the mutual exclusion were being managed by the Fischer-Lynch test-and-set operator, which is considerably less restrictive than

Distributed Graph Recognition with Malicious Faultsa

{\bf PV}

Byzantine Firing Squad Using a Faulty External Source

-chunk.

Bounds on threshold dimension and disjoint threshold coverings (abstract only)

Bn threshold graphs, with B = 1.5A. Thus the difference between these two covering numbers can grow linearly in the number of vertices .

Mimimal threshold separators and memory requirements for synchronization (abstract only)

Given a graphG withn vertices andm edges, how many edges must be in the largest chordal subgraph ofG? Form=n2/4+1, the answer is 3n/2−1. Form=n2/3, it is 2n−3. Form=n2/3+1, it is at least 7n/3−6 and at most 8n/3−4. Similar questions are studied, with less complete results, for threshold graphs, interval graphs, and the stars on edges, triangles, andK4s.