5 research outputs found

    Independent sets and non-augmentable paths in generalizations of tournaments

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    AbstractWe study different classes of digraphs, which are generalizations of tournaments, to have the property of possessing a maximal independent set intersecting every non-augmentable path (in particular, every longest path). The classes are the arc-local tournament, quasi-transitive, locally in-semicomplete (out-semicomplete), and semicomplete k-partite digraphs. We present results on strongly internally and finally non-augmentable paths as well as a result that relates the degree of vertices and the length of longest paths. A short survey is included in the introduction

    Independent sets and non-augmentable paths in arc-locally in-semicomplete digraphs and quasi-arc-transitive digraphs

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    AbstractA digraph is arc-locally in-semicomplete if for any pair of adjacent vertices x,y, every in-neighbor of x and every in-neighbor of y either are adjacent or are the same vertex. A digraph is quasi-arc-transitive if for any arc xy, every in-neighbor of x and every out-neighbor of y either are adjacent or are the same vertex. Laborde, Payan and Xuong proposed the following conjecture: Every digraph has an independent set intersecting every non-augmentable path (in particular, every longest path). In this paper, we shall prove that this conjecture is true for arc-locally in-semicomplete digraphs and quasi-arc-transitive digraphs

    On the existence and number of (k+1)(k+1)-kings in kk-quasi-transitive digraphs

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    Let D=(V(D),A(D))D=(V(D), A(D)) be a digraph and k≥2k \ge 2 an integer. We say that DD is kk-quasi-transitive if for every directed path (v0,v1,...,vk)(v_0, v_1,..., v_k) in DD, then (v0,vk)∈A(D)(v_0, v_k) \in A(D) or (vk,v0)∈A(D)(v_k, v_0) \in A(D). Clearly, a 2-quasi-transitive digraph is a quasi-transitive digraph in the usual sense. Bang-Jensen and Gutin proved that a quasi-transitive digraph DD has a 3-king if and only if DD has a unique initial strong component and, if DD has a 3-king and the unique initial strong component of DD has at least three vertices, then DD has at least three 3-kings. In this paper we prove the following generalization: A kk-quasi-transitive digraph DD has a (k+1)(k+1)-king if and only if DD has a unique initial strong component, and if DD has a (k+1)(k+1)-king then, either all the vertices of the unique initial strong components are (k+1)(k+1)-kings or the number of (k+1)(k+1)-kings in DD is at least (k+2)(k+2).Comment: 17 page
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