In [19] Huang gave a characterization of local tournaments. His characterization involves arc-reversals and therefore may not be easily used to solve other structural problems on locally semicomplete digraphs (where one deals with a fixed locally semicomplete digraph). In this paper we derive a classification of locally semicomplete digraphs which is very useful for studying structural properties of locally semicomplete digraphs and which does not depend on Huang's characterization. An advantage of this new classification of locally semicomplete digraphs is that it allows one to prove results for locally semicomplete digraphs without reproving the same statement for tournaments. We use our result to characterize pancyclic and vertex pancyclic locally semicomplete digraphs and to show the existence of a polynomial algorithm to decide whether a given locally semicomplete digraph has a kernel.
It is shown that every k-connected locally semicomplete digraph D with minimum outdegree at least 2k and minimum indegree at least 2k -2 has at least rn = max(2, k} vertices XI, x2, . . . , x, such that Dx; is k-connected for i 1,2,. . . , m.
In this article, we show that every bridgeless graph G of order n and maximum degree Δ has an orientation of diameter at most n−Δ+3. We then use this result and the definition NGfalse(Hfalse)=⋃v∈V(H)NGfalse(vfalse)∖Vfalse(Hfalse), for every subgraph H of G, to give better bounds in the case that G contains certain clusters of high‐degree vertices, namely: For every edge e, G has an orientation of diameter at most n−false|NG(e)false|+4, if e is on a triangle and at most n−false|NG(e)false|+5, otherwise. Furthermore, for every bridgeless subgraph H of G, there is such an orientation of diameter at most n−false|NG(H)false|+3. Finally, if G is bipartite, then we show the existence of an orientation of diameter at most 2(|A|− deg Gfalse(sfalse))+7, for every partite set A of G and s∈V(G)∖A. This particularly implies that balanced bipartite graphs have an orientation of diameter at most n−2Δ+7. For each bound, we give a polynomial‐time algorithm to construct a corresponding orientation and an infinite family of graphs for which the bound is sharp.
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