Semicomplete Multipartite Digraphs Research Articles

Disjoint Paths in Decomposable Digraphs

AbstractThe 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].

Strongly quasi-Hamiltonian-connected semicomplete multipartite digraphs

A semicomplete multipartite or semicomplete c-partite digraph D is a biorientation of a c-partite graph. A semicomplete multipartite digraph D is called strongly quasi-Hamiltonian-connected, if for any two distinct vertices x and y of D, there is a path P from x to y such that P contains at least one vertex from each partite set of D.In this paper we show that every 4-strong semicomplete multipartite digraph is strongly quasi-Hamiltonian-connected. This generalizes a result due to Thomassen for tournaments.

Quasi-hamiltonian paths in semicomplete multipartite digraphs

A quasi-hamiltonian path in a semicomplete multipartite digraph D is a path which visits each maximal independent set (also called a partite set) of D at least once. This is a generalization of a hamiltonian path in a tournament.In this paper we investigate the complexity of finding a quasi-hamiltonian path, in a given semicomplete multipartite digraph, from a prescribed vertex x to a prescribed vertex y as well as the complexity of finding a quasi-hamiltonian path whose end vertices belong to a given set of two vertices {x,y}. While both of these problems are polynomially solvable for semicomplete digraphs (here all maximal independent sets have size one), we prove that the first problem is NP-complete and describe a polynomial algorithm for the latter problem.

The complexity of the minimum cost homomorphism problem for semicomplete digraphs with possible loops

For digraphs D and H , a mapping f : V ( D ) → V ( H ) is a homomorphism of D to H if u v ∈ A ( D ) implies f ( u ) f ( v ) ∈ A ( H ) . For a fixed digraph H , the homomorphism problem is to decide whether an input digraph D admits a homomorphism to H or not, and is denoted as HOM( H ). An optimization version of the homomorphism problem was motivated by a real-world problem in defence logistics and was introduced in Gutin, Rafiey, Yeo and Tso (2006) [13]. If each vertex u ∈ V ( D ) is associated with costs c i ( u ) , i ∈ V ( H ) , then the cost of the homomorphism f is ∑ u ∈ V ( D ) c f ( u ) ( u ) . For each fixed digraph H , we have the minimum cost homomorphism problem for H and denote it as MinHOM( H ). The problem is to decide, for an input graph D with costs c i ( u ) , u ∈ V ( D ) , i ∈ V ( H ) , whether there exists a homomorphism of D to H and, if one exists, to find one of minimum cost. Although a complete dichotomy classification of the complexity of MinHOM( H ) for a digraph H remains an unsolved problem, complete dichotomy classifications for MinHOM( H ) were proved when H is a semicomplete digraph Gutin, Rafiey and Yeo (2006) [10], and a semicomplete multipartite digraph Gutin, Rafiey and Yeo (2008) [12,11]. In these studies, it is assumed that the digraph H is loopless. In this paper, we present a full dichotomy classification for semicomplete digraphs with possible loops, which solves a problem in Gutin and Kim (2008) [9].

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 K 4 ( 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 K 4 ( Q ) ⊆ K 4 ( T ) , Tan [On the kings and kings-of-kings in semicomplete multipartite digraphs, Discrete Math. 290 (2005) 249–258] showed that K 3 ( Q ) ⊆ K 3 ( T ) and K 2 ( Q ) = K 2 ( 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 K 3 ( Q ) need not be the same as K 3 ( T ) in general and posed the problem: characterize all those semicomplete multipartite digraphs T such that K 3 ( Q ) = K 3 ( T ) . In the course of proving our result (1), we (2) show that K 3 ( Q ) = K 3 ( T ) for all semicomplete multipartite digraphs T with no transmitters such that Q is a semicomplete digraph.

On the kings and kings-of-kings in semicomplete multipartite digraphs

Koh and Tan showed in (Evaluation of the number of kings in a multipartite tournament, submitted for publication.) that the subdigraph induced by the 4-kings of an n-partite tournament with no transmitters, where n ⩾ 3 , contains no transmitters. We extend this result to the class of semicomplete n-partite digraph, where n ⩾ 2 . 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 K 4 ( T ) . In this paper, we (1) show that Q has no transmitters, (2) obtain some results on the 2-kings, 3-kings and 4-kings in T. While it is trivial that K 4 ( Q ) ⊆ K 4 ( T ) , we further prove that (3) K 3 ( Q ) ⊆ K 3 ( T ) and (4) K 2 ( Q ) = K 2 ( T ) . Maurer (Math. Mag. 53 (1980) 67) introduced the concept of kings-of-kings in tournaments. Following Maurer, we investigate the r-kings-of-kings of semicomplete multipartite digraphs with no transmitters for r = 1 , 2 , 3 , 4 . Some problems on the r-kings-of-kings are posed.

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), and one of the partite sets of it consists of a single vertex, say v ,t henD 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 ,t henD has 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.

Longest paths through an arc in strong semicomplete multipartite digraphs

A digraph obtained by replacing each edge of a complete n-partite graph by an arc or a pair of mutually opposite arcs is called a semicomplete n-partite digraph. An n-partite tournament is an orientation of a complete n-partite graph. If D is a strongly connected semicomplete n-partite digraph, then we prove that every arc of D which belongs to a directed cycle of length at least 3, is contained in a directed path of order ⌈( n+3)/2⌉. Consequently, every arc of a strongly connected n-partite tournament is contained in a directed path with ⌈( n+3)/2⌉ vertices. This bound is new even for tournaments. If in addition, every partite set consists of at least two vertices, then we even have the slightly better lower bound ⌈( n+5)/2⌉. Various families of examples will show that these results are best possible.

The Minimum Spanning Strong Subdigraph Problem for Extended Semicomplete Digraphs and Semicomplete Bipartite Digraphs

We consider the problem (minimum spanning strong subdigraph (MSSS)) of finding the minimum number of arcs in a spanning strongly connected subdigraph of a strongly connected digraph. This problem is NP-hard for general digraphs since it generalizes the Hamiltonian cycle problem. We characterize the number of arcs in a minimum spanning strong subdigraph for digraphs which are either extended semicomplete or semicomplete bipartite. Our proofs lead to polynomial algorithms for finding an optimal subdigraph for every digraph from each of these classes. Our proofs are based on a number of results (some of which are new and interesting in their own right) on the structure of cycles and paths in these graphs. Recently, it was shown that the Hamiltonian cycle problem is polynomially solvable for semicomplete multipartite digraphs, a superclass of the two classes above [15]. We conjecture that the MSSS problem is also polynomial for this class of digraphs.

Solution of a Conjecture of Volkmann on the Number of Vertices in Longest Paths and Cycles of Strong Semicomplete Multipartite Digraphs

A digraph obtained by replacing each edge of a complete multipartite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete multipartite digraph. L. Volkmann conjectured that l≤2c−1, where l (c, respectively) is the number of vertices in a longest path (longest cycle) of a strong semicomplete multipartite digraph. The bound on l is sharp. We settle this conjecture in affirmative.

The ratio of the longest cycle and longest path in semicomplete multipartite digraphs

A digraph obtained by replacing each edge of a complete n-partite graph by an arc or a pair of mutually opposite arcs is called a semicomplete n-partite digraph. We call α( D)=max 1⩽ i⩽ n {| V i |} the independence number of the semicomplete n-partite digraph D, where V 1, V 2,…, V n are the partite sets of D. Let p and c, respectively, denote the number of vertices in a longest directed path and the number of vertices in a longest directed cycle of a digraph D. Recently, Gutin and Yeo proved that c⩾( p+1)/2 for every strongly connected semicomplete n-partite digraph D. In this paper we present for the special class of semicomplete n-partite digraphs D with connectivity κ( D)= α( D)−1⩾1 the better bound c⩾ κ(D) κ(D)+1 (p+1). In addition, we present examples which show that this bound is best possible.

Longest paths in strong spanning oriented subgraphs of strong semicomplete multipartite digraphs

A digraph obtained by replacing each edge of a complete multipartite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete multipartite digraph. Volkmann (Manuscript, RWTH Aachen, Germany, June 1998) raised the following question: Let D be a strong semicomplete multipartite digraph with a longest path of length l. Does there exist a strong spanning oriented subgraph of D with a longest path of length l? We provide examples which show that the answer to this question is negative. We also demonstrate that every strong semicomplete multipartite digraph D, which is not bipartite with a partite set of cardinality one, has a strong spanning oriented subgraph of D with a longest path of length at least l−2. This bound is sharp.

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

Sufficient conditions for semicomplete multipartite digraphs to be Hamiltonian

A semicomplete multipartite digraph is obtained by replacing each edge of a complete multipartite graph by an arc or by a pair of two mutually opposite arcs. Very recently, Yeo (J. Graph Theory 24 (1997) 175–185), proved that every regular semicomplete multipartite digraph is Hamiltonian. With this, Yeo confirmed a conjecture of Zhang (Ann. Discrete Math. 41 (1989) 499–514). In the first part of this paper, a generalization of regularity is considered. We extend Yeo's result to semicomplete multipartite digraphs that satisfy this generalized condition apart from exactly two exceptions. In the second part, we introduce the so-called semi-partition complete digraphs and show that this family is Hamiltonian or cycle complementary, when, clearly, the cardinality of each partite set is less than or equal to half the order.

How Close to Regular Must a Semicomplete Multipartite Digraph Be to Secure Hamiltonicity?

Let $D$ be a semicomplete multipartite digraph, with partite sets (called color classes) $V_1,V_2,\ldots,V_c$, such that $|V_1| \leq |V_2| \leq \ldots \leq |V_c|$. Define $\FF{D}=|V(D)|-3|V_c|+1$ and $\GG{D}=\frac{|V(D)|-|V_{c-1}|-2|V_c|+2}{2}$. We define the irregularity $i(D)$ of $D$ to be $\max |d^+(x)-d^-(y)|$ over all vertices $x$ and $y$ of $D$ (possibly $x=y$). We define the local irregularity $\Iv{D}$ of $D$ to be $\max|d^+(x)-d^-(x)|$ over all vertices $x$ of $D$ and we define the global irregularity of $D$ to be $\ig{D}= max\{d^+(x),d^-(x) : x\in V(D)\}- min\{d^+(y),d^-(y) : y\in V(D)\}$. In this paper we show that if $\ig{D} \leq \GG{D}$ or if $\Iv{D} \leq \MinFG{D}$ then $D$ is Hamiltonian. We furthermore show how this implies a theorem which generalizes two results by Volkmann and solves a stated problem and a conjecture from \cite{Volk}. Our result also gives support to the conjecture from \cite{Volk} that all diregular $c$-partite tournaments ($c \geq 4$) are pancyclic. Finally we show that our result in some sense is best possible, by giving an infinite class of non-Hamiltonian semicomplete multipartite digraphs, $D$, with $\ig{D}=i(D)=\Iv{D}=\GG{D}+\frac{1}{2} \leq \FF{D}+1$.

A Polynomial Time Algorithm for Finding a Cycle Covering a Given Set of Vertices in a Semicomplete Multipartite Digraph

The existence of a polynomial time algorithm for finding a cycle covering a given set of vertices in a semicomplete multipartite digraph (if it exists) was conjectured by J. Bang-Jensen, G. Gutin, and A. Yeo. The analog problem for semicomplete bipartite digraphs was conjectured by J. Bang-Jensen and Y. Manoussakis. We prove the conjecture of Bang-Jensen et al. in the affirmative, which also implies the second conjecture. We, furthermore, give a polynomial time algorithm to find a cycle covering a given set of vertices in a semicomplete bipartite digraph, which is the longest of all such cycles, and we conjecture that such an algorithm also exists for semicomplete multipartite digraphs.

Outpaths in semicomplete multipartite digraphs

An outpath of a vertex x (an arc xy, respectively) in a digraph is a directed path starting at x ( xy, respectively) such that x dominates the endvertex of the path only if the endvertex also dominates x. Firstly, we show that if D is a strongly connected semicomplete n-partite ( n⩾3) digraph, then every vertex v of D has an outpath of length k−1 for all k∈{3,4,…, n}. Our result generalizes a theorem of Moon (Canad. Math. Bull. 9 (1966) 297–301) for tournaments. Secondly, we show that if T is a regular n-partite ( n⩾3) tournament, then every arc of T has an outpath of length k−1 for all k satisfying 3⩽ k⩽ n. This result extends a theorem of Alspach (Canad. Math. Bull. 10 (1967) 283–286) for regular tournaments to regular multipartite tournaments.

A polynomial algorithm for the Hamiltonian cycle problem in semicomplete multipartite digraphs

We describe a polynomial algorithm for the Hamiltonian cycle problem for semicomplete multipartite digraphs. The existence of such an algorithm was conjectured in \cite{gthesis} (see also \cite{sur}).

A Polynomial Algorithm for the Hamiltonian Cycle Problem in Semicomplete Multipartite Digraphs

We describe a polynomial algorithm for the Hamiltonian cycle problem for semicomplete multipartite digraphs. The existence of such an algorithm was conjectured in G. Gutin, Paths and cycles in digraphs. Ph. D. thesis, Tel Aviv Univ., 1993. (see also G. Gutin, J Graph Theory 19 (1995) 481–505). © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 111–132, 1998

One-diregular subgraphs in semicomplete multipartite digraphs

The problem of finding necessary and sufficient conditions for a semicomplete multipartite digraph (SMD) to be Hamiltonian, seems to be both very interesting and difficult. Bang-Jensen, Gutin and Huang (Discrete Math to appear) proved a sufficient condition for a SMD to be Hamiltonian. A strengthening of this condition, shown in this paper, allows us to prove the following three results. We prove that every k-strong SMD with at most k-vertices in each color class is Hamiltonian and every k-strong SMD has a cycle through any set of k vertices. These two statements were stated as conjectures by Volkmann (L. Volkmann, a talk at the second Krakw Conference of Graph Theory (1994)) and Bang-Jensen, Gutin, and Yeo (J. Bang-Jensen, G. Gutin, and A. Yeo, On k-strong and k-cyclic digraphs, submitted), respectively. We also prove that every diregular SMD is Hamiltonian, which was conjectured in a weaker form by Zhang (C.-Q. Zhang, Hamilton paths in multipartite oriented graphs, Ann Discrete Math. 41 (1989), 499–581). © 1997 John Wiley & Sons, Inc.