Minimum Cost Homomorphisms to Locally Semicomplete and Quasi-Transitive Digraphs

For digraphs $G$ and $H$, a homomorphism of $G$ to $H$ is a mapping $f:\ V(G)\dom V(H)$such that$uv\in A(G)$implies$f(u)f(v)\in A(H)$. If, moreover, each vertex$u \in V(G)$is associated with costs$c_i(u), i \in V(H)$, then the cost of a homomorphism$f$is$\sum_{u\in V(G)}c_{f(u)}(u)$. For each fixed digraph$H$, the minimum cost homomorphism problem for$H$, denoted MinHOM($H$), can be formulated as follows: Given an input digraph$G$, together with costs$c_i(u)$,$u\in V(G)$,$i\in V(H)$, decide whether there exists a homomorphism of$G$to$H$ and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems such as the minimum cost chromatic partition and repair analysis problems. We focus on the minimum cost homomorphism problem for locally semicomplete digraphs and quasi-transitive digraphs which are two well-known generalizations of tournaments. Using graph-theoretic characterization results for the two digraph classes, we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for both classes

Hamiltonian paths containing a given arc, in almost regular bipartite tournaments

AbstractA tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph. If x is a vertex of a digraph D, then we denote by d+(x) and d−(x) the outdegree and indegree of x, respectively. The global irregularity of a digraph D is defined by ig(D)=max{d+(x),d−(x)}−min{d+(y),d−(y)} over all vertices x and y of D (including x=y). If ig(D)⩽1, then D is called almost regular, and if ig(D)=0, then D is regular.More than 10 years ago, Amar and Manoussakis and independently Wang proved that every arc of a regular bipartite tournament is contained in a directed Hamiltonian cycle. In this paper, we prove that every arc of an almost regular bipartite tournament T is contained in a directed Hamiltonian path if and only if the cardinalities of the partite sets differ by at most one and T is not isomorphic to T3,3, where T3,3 is an almost regular bipartite tournament with three vertices in each partite set.As an application of this theorem and other results, we show that every arc of an almost regular c-partite tournament D with the partite sets V1,V2,…,Vc such that |V1|=|V2|=⋯=|Vc|, is contained in a directed Hamiltonian path if and only if D is not isomorphic to T3,3

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

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