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On partially directed p-graphs. (English) Zbl 0608.05037
In the following G may be assumed to be a simple digraph [by results of J. Bosák, Theor. Appl. Graphs, Proc. Kalamazoo 1976, Lect. Notes Math. 642, 75-85 (1978; Zbl 0371.05016); Math. Slovaca 29, 181-196 (1979; Zbl 0407.05057)]. G is said to be homogeneous of valency d if $$\text{indegree}(v) = \text{outdegree}(v) = d$$ for every vertex v of G. G is a P-graph if for every ordered pair u,v of vertices of G there exists exactly one path from u to v such that the length of the path does not exceed the diameter of G. G is a quasitree if for every ordered pair u,v of vertices of G there exists exactly one path from u to v. G is a T- graph if for every ordered pair u,v of vertices of G there exists exactly one trail from u to v such that the length of the trail does not exceed the diameter of G. A graph of type P(d,k) is a homogeneous P-graph theory of valency d and with a finite diameter k. The main results of the author are as follows. (1) Every P-graph is either a quasitree or a homogeneous block $$(=connected$$ graph without a cutpoint) with a finite diameter. (2) Every P-graph with diameter at most two is a T-graph. At the end the author raises the following question: For which integers d and k does there exist a graph of type $$P(d,k)$$?
##### MSC:
 05C20 Directed graphs (digraphs), tournaments
##### Keywords:
digraph; P-graph; quasitree; T-graph
##### References:
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