1,827 research outputs found
All Minimal Prime Extensions of Hereditary Classes of Graphs
The substitution composition of two disjoint graphs G1 and G2 is obtained by first removing a vertex x from G2 and then making every vertex in G1 adjacent to all neighbours of x in G2. Let F be a family of graphs defined by a set Z* of forbidden configurations. Giakoumakis [V. Giakoumakis, On the closure of graphs under substitution, Discrete Mathematics 177 (1997) 83–97] proved that F∗, the closure under substitution of F, can be characterized by a set Z∗ of forbidden configurations — the minimal prime extensions of Z. He also showed that Z∗ is not necessarily a finite set. Since substitution preserves many of the properties of the composed graphs, an important problem is the following: find necessary and sufficient conditions for the finiteness of Z∗. Giakoumakis [V. Giakoumakis, On the closure of graphs under substitution, Discrete Mathematics 177 (1997) 83–97] presented a sufficient condition for the finiteness of Z∗ and a simple method for enumerating all its elements. Since then, many other researchers have studied various classes of graphs for which the substitution closure can be characterized by a finite set of forbidden configurations.
The main contribution of this paper is to completely solve the above problem by characterizing all classes of graphs having a finite number of minimal prime extensions. We then go on to point out a simple way for generating an infinite number of minimal prime extensions for all the other classes of F∗
Classification of graph C*-algebras with no more than four primitive ideals
We describe the status quo of the classification problem of graph C*-algebras
with four primitive ideals or less
Pairs of orthogonal countable ordinals
We characterize pairs of orthogonal countable ordinals. Two ordinals
and are orthogonal if there are two linear orders and on the
same set with order types and respectively such that the
only maps preserving both orders are the constant maps and the identity map. We
prove that if and are two countable ordinals, with , then and are orthogonal if and only if either
or and
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