Kings in semicomplete multipartite digraphs

A digraph obtained by replacing each edge of a complete p‐partite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete p‐partite digraph, or just a semicomplete multipartite digraph. A semicomplete multipartite digraph with no cycle of length two is a multipartite tournament. In a digraph D, an r‐king is a vertex q such that every vertex in D can be reached from q by a path of length at most r. Strengthening a theorem by K. M. Koh and B. P. Tan (Discr Math 147 (1995), 171–183) on the number of 4‐kings in multipartite tournaments, we characterize semicomplete multipartite digraphs, which have exactly k 4‐kings for every k = 1, 2, 3, 4, 5. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 177‐183, 2000     

Arc‐Disjoint In‐ and Out‐Branchings With the Same Root in Locally Semicomplete Digraphs

Deciding whether a digraph contains a pair of arc‐disjoint in‐ and out‐branchings rooted at a specified vertex is a well‐known NP‐complete problem (as proved by Thomassen, see 2 ). This problem has been shown to be polynomial time solvable for semicomplete digraphs 2 and for quasi‐transitive digraphs 6 . In this article, we study the problem for locally semicomplete digraphs. We characterize locally semicomplete digraphs that contain a pair of arc‐disjoint in‐ and out‐branchings rooted at a specified vertex. Our proofs are constructive and imply the existence of a polynomial time algorithm for finding the desired branchings when they exist. Our results generalizes those from 2 for semicomplete digraphs and solves an open problem from 4 .     

Locally semicomplete digraphs: A generalization of tournaments

In this paper we introduce a new class of directed graphs called locally semicomplete digraphs. These are defined to be those digraphs for which the following holds: for every vertex x the vertices dominated by x induce a semicomplete digraph and the vertices that dominate x induce a semicomplete digraph. (A digraph is semicomplete if for any two distinct vertices u and ν, there is at least one arc between them.) This class contains the class of semicomplete digraphs, but is much more general. In fact, the class of underlying graphs of the locally semi-complete digraphs is precisely the class of proper circular-arc graphs (see [13], Theorem 3). We show that many of the classic theorems for tournaments have natural analogues for locally semicomplete digraphs. For example, every locally semicomplete digraph has a directed Hamiltonian path and every strong locally semicomplete digraph has a Hamiltonian cycle. We also consider connectivity properties, domination orientability, and algorithmic aspects of locally semicomplete digraphs. Some of the results on connectivity are new, even when restricted to semicomplete digraphs.     

Disjoint quasi‐kernels in digraphs

A quasi‐kernel in a digraph is an independent set of vertices such that any vertex in the digraph can reach some vertex in the set via a directed path of length at most two. Chvátal and Lovász proved that every digraph has a quasi‐kernel. Recently, Gutin et al. raised the question of which digraphs have a pair of disjoint quasi‐kernels. Clearly, a digraph has a pair of disjoint quasi‐kernels cannot contain sinks, that is, vertices of outdegree zero, as each such vertex is necessarily included in a quasi‐kernel. However, there exist digraphs which contain neither sinks nor a pair of disjoint quasi‐kernels. Thus, containing no sinks is not sufficient in general for a digraph to have a pair of disjoint quasi‐kernels. In contrast, we prove that, for several classes of digraphs, the condition of containing no sinks guarantees the existence of a pair of disjoint quasi‐kernels. The classes contain semicomplete multipartite, quasi‐transitive, and locally semicomplete digraphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:251‐260, 2008     

ON THE EXPONENT SET OF PRIMITIVELOCALLY SEMICOMPLETE DIGRAPHS

A locally semicomplete digraph is a digraph D=(V,A) satisfying the following condi-tion for every vertex x∈V the D[O(x)] and D[I(x)] are semicomplete digraphs. In this paper,we get some properties of cycles and determine the exponent set of primitive locally semicompleted digraphs.     

c-Pancyclic Partial Ordering and （c-1）-Pan-Outpath Partial Ordering in Semicomplete Multipartite Digraphs

An outpath of a vertex v in a digraph is a path starting at v such that v dominates the end vertex of the path only if the end vertex also dominates v.First we show that letting D be a strongly connected semicomplete c-partite digraph (c≥3)1 and one of the partite sets of it consists of a single vertex, say v, then D has a c-pancyclic partial ordering from v, which generalizes a result about pancyclicity of multipartite tournaments obtained by Gutin in 1993.Then we prove that letting D be a strongly connected semicomplete c-partite digraph with c≥3 and letting v be a vertex of D,then Dhas a(c-1)-pan-outpath partly ordering from v.This result improves a theorem about outpaths in semicomplete multipartite digraphs obtained by Guo in 1999.     

A Characterization of Locally Semicomplete CKI-digraphs

A digraph is locally-in semicomplete if for every vertex of D its in-neighborhood induces a semicomplete digraph and it is locally semicomplete if for every vertex of D the in-neighborhood and the out-neighborhood induces a semicomplete digraph. The locally semicomplete digraphs where characterized in 1997 by Bang-Jensen et al. and in 1998 Bang-Jensen and Gutin posed the problem if finding a kernel in a locally-in semicomplete digraph is polynomial or not. A kernel of a digraph is a set of vertices, which is independent and absorbent. A digraph D such that every proper induced subdigraph of D has a kernel is said to be critical kernel imperfect digraph (CKI-digraph) if the digraph D does not have a kernel. A digraph without an induced CKI-digraph as a subdigraph does have a kernel. We characterize the locally semicomplete digraphs, which are CKI. As a consequence of this characterization we conclude that determinate whether a locally semicomplete digraph is a CKI-digraph or not, is polynomial.     

Finding cheapest cycles in vertex-weighted quasi-transitive and extended semicomplete digraphs

We consider the problem of finding a minimum cost cycle in a digraph with real-valued costs on the vertices. This problem generalizes the problem of finding a longest cycle and hence is NP-hard for general digraphs. We prove that the problem is solvable in polynomial time for extended semicomplete digraphs and for quasi-transitive digraphs, thereby generalizing a number of previous results on these classes. As a byproduct of our method we develop polynomial algorithms for the following problem: Given a quasi-transitive digraph D with real-valued vertex costs, find, for each j=1,2,…,|V(D)|, j disjoint paths P₁,P₂,…,P_j such that the total cost of these paths is minimum among all collections of j disjoint paths in D.     

Spanning 2-strong tournaments in 3-strong semicomplete digraphs

We prove that every 3-strong semicomplete digraph on at least 5 vertices contains a spanning 2-strong tournament. Our proof is constructive and implies a polynomial algorithm for finding a spanning 2-strong tournament in a given 3-strong semicomplete digraph. We also show that there are infinitely many (2k−2)-strong semicomplete digraphs which contain no spanning k-strong tournament and conjecture that every(2k−1)-strong semicomplete digraph which is not the complete digraph on 2k vertices contains a spanning k-strong tournament.     

On semicomplete multipartite digraphs whose king sets are semicomplete digraphs

Reid [Every vertex a king, Discrete Math. 38 (1982) 93-98] showed that a non-trivial tournament H is contained in a tournament whose 2-kings are exactly the vertices of H if and only if H contains no transmitter. Let T be a semicomplete multipartite digraph with no transmitters and let K_r(T) denote the set of r-kings of T. Let Q be the subdigraph of T induced by K₄(T). Very recently, Tan [On the kings and kings-of-kings in semicomplete multipartite digraphs, Discrete Math. 290 (2005) 249-258] proved that Q contains no transmitters and gave an example to show that the direct extension of Reid's result to semicomplete multipartite digraphs with 2-kings replaced by 4-kings is not true. In this paper, we (1) characterize all semicomplete digraphs D which are contained in a semicomplete multipartite digraph whose 4-kings are exactly the vertices of D. While it is trivial that K₄(Q)⊆K₄(T), Tan [On the kings and kings-of-kings in semicomplete multipartite digraphs, Discrete Math. 290 (2005) 249-258] showed that K₃(Q)⊆K₃(T) and K₂(Q)=K₂(T). Tan [On the kings and kings-of-kings in semicomplete multipartite digraphs, Discrete Math. 290 (2005) 249-258] also provided an example to show that K₃(Q) need not be the same as K₃(T) in general and posed the problem: characterize all those semicomplete multipartite digraphs T such that K₃(Q)=K₃(T). In the course of proving our result (1), we (2) show that K₃(Q)=K₃(T) for all semicomplete multipartite digraphs T with no transmitters such that Q is a semicomplete digraph.     

On the structure of strong 3-quasi-transitive digraphs

In this paper, D=(V(D),A(D)) denotes a loopless directed graph (digraph) with at most one arc from u to v for every pair of vertices u and v of V(D). Given a digraph D, we say that D is 3-quasi-transitive if, whenever u→v→w→z in D, then u and z are adjacent or u=z. In Bang-Jensen (2004) [3], Bang-Jensen introduced 3-quasi-transitive digraphs and claimed that the only strong 3-quasi-transitive digraphs are the strong semicomplete digraphs and strong semicomplete bipartite digraphs. In this paper, we exhibit a family of strong 3-quasi-transitive digraphs distinct from strong semicomplete digraphs and strong semicomplete bipartite digraphs and provide a complete characterization of strong 3-quasi-transitive digraphs.     

Disjoint Paths in Decomposable Digraphs

The k‐linkage problem is as follows: given a digraph and a collection of k terminal pairs such that all these vertices are distinct; decide whether D has a collection of vertex disjoint paths such that is from to for . A digraph is k‐linked if it has a k‐linkage for every choice of 2k distinct vertices and every choice of k pairs as above. The k‐linkage problem is NP‐complete already for [11] and there exists no function such that every ‐strong digraph has a k‐linkage for every choice of 2k distinct vertices of D [17]. Recently, Chudnovsky et al. [9] gave a polynomial algorithm for the k‐linkage problem for any fixed k in (a generalization of) semicomplete multipartite digraphs. In this article, we use their result as well as the classical polynomial algorithm for the case of acyclic digraphs by Fortune et al. [11] to develop polynomial algorithms for the k‐linkage problem in locally semicomplete digraphs and several classes of decomposable digraphs, including quasi‐transitive digraphs and directed cographs. We also prove that the necessary condition of being ‐strong is also sufficient for round‐decomposable digraphs to be k‐linked, obtaining thus a best possible bound that improves a previous one of . Finally we settle a conjecture from [3] by proving that every 5‐strong locally semicomplete digraph is 2‐linked. This bound is also best possible (already for tournaments) [1].     

The diameter of randomly perturbed digraphs and some applications

The central observation of this paper is that if εn random arcs are added to any n‐node strongly connected digraph with bounded degree then the resulting graph has diameter 𝒪(lnn) with high probability. We apply this to smoothed analysis of algorithms and property testing. Smoothed Analysis: Recognizing strongly connected digraphs is a basic computational task in graph theory. Even for digraphs with bounded degree, it is NL‐complete. By XORing an arbitrary bounded degree digraph with a sparse random digraph R ∼ 𝔻_n,ε/n we obtain a “smoothed” instance. We show that, with high probability, a log‐space algorithm will correctly determine if a smoothed instance is strongly connected. We also show that if NL ⫅̸ almost‐L then no heuristic can recognize similarly perturbed instances of (s,t)‐connectivity. Property Testing: A digraph is called k‐linked if, for every choice of 2k distinct vertices s₁,…,s_k,t₁,…,t_k, the graph contains k vertex disjoint paths joining s_r to t_r for r = 1,…,k. Recognizing k‐linked digraphs is NP‐complete for k ≥ 2. We describe a polynomial time algorithm for bounded degree digraphs, which accepts k‐linked graphs with high probability, and rejects all graphs that are at least εn arcs away from being k‐linked. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Efficient total domination in digraphs

We generalize the concept of efficient total domination from graphs to digraphs. An efficiently total dominating set X of a digraph D is a vertex subset such that every vertex of D has exactly one predecessor in X. We study graphs that permit an orientation having such a set and give complexity results and characterizations. Furthermore, we study the computational complexity of the (weighted) efficient total domination problem for several digraph classes. In particular we deal with most of the common generalizations of tournaments, like locally semicomplete and arc-locally semicomplete digraphs.     

Strongly Hamiltonian-connected locally semicomplete digraphs

We give some sufficient conditions for locally semicomplete digraphs to contain a hamiltonian path from a prescribed vertex to another prescribed vertex. As an immediate consequence of these, we obtain that every 4-connected locally semicomplete digraph is strongly hamiltonian-connected. Our results extend those of Thomassen [12] for tournaments. © 1996 John Wiley & Sons, Inc.     

List coloring digraphs

The dichromatic number of a digraph D is the least number k such that the vertex set of D can be partitioned into k parts each of which induces an acyclic subdigraph. Introduced by Neumann‐Lara in 1982, this digraph invariant shares many properties with the usual chromatic number of graphs and can be seen as the natural analog of the graph chromatic number. In this article, we study the list dichromatic number of digraphs, giving evidence that this notion generalizes the list chromatic number of graphs. We first prove that the list dichromatic number and the dichromatic number behave the same in many contexts, such as in small digraphs (by proving a directed version of Ohba's conjecture), tournaments, and random digraphs. We then consider bipartite digraphs, and show that their list dichromatic number can be as large as . We finally give a Brooks‐type upper bound on the list dichromatic number of digon‐free digraphs.     

Finding a subdivision of a prescribed digraph of order 4

The problem of when a given digraph contains a subdivision of a fixed digraph F is considered. Bang‐Jensen et al. [4] laid out foundations for approaching this problem from the algorithmic point of view. In this article, we give further support to several open conjectures and speculations about algorithmic complexity of finding F‐subdivisions. In particular, up to five exceptions, we completely classify for which 4‐vertex digraphs F, the F‐subdivision problem is polynomial‐time solvable and for which it is NP‐complete. While all NP‐hardness proofs are made by reduction from some version of the 2‐linkage problem in digraphs, some of the polynomial‐time solvable cases involve relatively complicated algorithms.     

On the structure of locally semicomplete digraphs

In this paper we continue the study, started by J. Bang-Jensen (1989), of locally semicomplete digraphs, a generalization of tournaments, to which many well-known tournament results extend. The underlying undirected graphs of the locally semicomplete digraphs are precisely the proper circular-arc graphs. We give new results on the structure of locally semicomplete digraphs, as well as several examples of properties of tournaments and semicomplete digraphs that do not extend to the class of locally semicomplete digraphs.     

Independent sets and non-augmentable paths in generalizations of tournaments

We 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

A 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.