A study on supereulerian digraphs and spanning trails in digraphs

A strong digraph D is eulerian if for any v ∈ V (D), d+D (v) = d−D (v). A digraph D is supereulerian if D contains a spanning eulerian subdigraph, or equivalently, a spanning closed directed trail. A digraph D is trailable if D has a spanning directed trail. This dissertation focuses on a study of trailable digraphs and supereulerian digraphs from the following aspects. 1. Strong Trail-Connected, Supereulerian and Trailable Digraphs. For a digraph D, D is trailable digraph if D has a spanning trail. A digraph D is strongly trail- connected if for any two vertices u and v of D, D posses both a spanning (u, v)-trail and a spanning (v,u)-trail. As the case when u = v is possible, every strongly trail-connected digraph is also su- pereulerian. Let D be a digraph. Let S(D) = {e ∈ A(D) : e is symmetric in D}. A digraph D is symmetric if A(D) = S(D). The symmetric core of D, denoted by J(D), has vertex set V (D) and arc set S(D). We have found a well-characterized digraph family D each of whose members does not have a spanning trail with its underlying graph spanned by a K2,n−2 such that for any strong digraph D with its matching number α′(D) and arc-strong-connectivity λ(D), if n = |V (D)| ≥ 3 and λ(D) ≥ α′(D) − 1, then each of the following holds. (i) There exists a family D of well-characterized digraphs such that for any digraph D with α′(D) ≤ 2, D has a spanning trial if and only if D is not a member in D. (ii) If α′(D) ≥ 3, then D has a spanning trail. (iii) If α′(D) ≥ 3 and n ≥ 2α′(D) + 3, then D is supereulerian. (iv) If λ(D) ≥ α′(D) ≥ 4 and n ≥ 2α′(D) + 3, then for any pair of vertices u and v of D, D contains a spanning (u, v)-trail. 2. Supereulerian Digraph Strong Products. A cycle vertex cover of a digraph D is a collection of directed cycles in D such that every vertex in D lies in at least one dicycle in this collection, and such that the union of the arc sets of these directed cycles induce a connected subdigraph of D. A subdigraph F of a digraph D is a circulation if for every vertex v in F, the indegree of v equals its outdegree, and a spanning circulation if F is a cycle factor. Define f(D) to be the smallest cardinality of a cycle vertex cover of the digraph D/F obtained from D by contracting all arcs in F , among all circulations F of D. In [International Journal of Engineering Science Invention, 8 (2019) 12-19], it is proved that if D1 and D2 are nontrivial strong digraphs such that D1 is supereulerian and D2 has a cycle vertex cover C′ with |C′| ≤ |V (D1)|, then the Cartesian product D1 and D2 is also supereulerian. We prove that for strong digraphs D1 and D2, if for some cycle factor F1 of D1, the digraph formed from D1 by contracting arcs in F1 is hamiltonian with f(D2) not bigger than |V (D1)|, then the strong product D1 and D2 is supereulerian

Independent sets and non-augmentable paths in generalizations of tournaments

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

Packing Strong Subgraph in Digraphs

In this paper, we study two types of strong subgraph packing problems in digraphs, including internally disjoint strong subgraph packing problem and arc-disjoint strong subgraph packing problem. These problems can be viewed as generalizations of the famous Steiner tree packing problem and are closely related to the strong arc decomposition problem. We first prove the NP-completeness for the internally disjoint strong subgraph packing problem restricted to symmetric digraphs and Eulerian digraphs. Then we get inapproximability results for the arc-disjoint strong subgraph packing problem and the internally disjoint strong subgraph packing problem. Finally we study the arc-disjoint strong subgraph packing problem restricted to digraph compositions and obtain some algorithmic results by utilizing the structural properties

Vertex heaviest paths and cycles in quasi-transitive digraphs.

A digraph D is called a quasi-transitive digraph (QTD) if for any triple x,y,z of distinct vertices of D such that (x, y) and (y, z) are arcs of D there is at least one arc from x and z or from z to x. Solving a conjecture by Bang-Jensen and Huang (1995), Gutin (1995) described polynomial algorithms for finding a Hamiltonian cycle and a Hamiltonian path (if it exists) in a QTD. The approach taken in that paper cannot be used to find a longest path or cycle in polynomial time. We present a principally new approach that leads to polynomial algorithms for finding vertex heaviest paths and cycles in QTDs with non-negative weights on the vertices. This, in particular, provides an answer to a question by N. Alon on longest paths and cycles in QTDs