Restricted arc-connectivity of generalized tournaments

If D is a strongly connected digraph, then an arc set S of D is called a restricted arc-cut of D if D − S has a non-trivial strong component D1 such that D − V (D1) contains an arc. Recently, Volkmann [12] defined the restricted arc-connectivity λ(D) as the minimum cardinality over all restricted arc-cuts S. A strongly connected digraph D is called λconnected when λ(D) exists. Let k ≥ 2 be an integer. An arc set S of D is a k-restricted arc-cut of D if D − S contains at least k non-trivial strong components. Volkmann [Inform. Process. Lett. 103 (2007), 234– 239] also defined the k-restricted arc-connectivity λ′k(D) as the minimum cardinality over all k-restricted arc-cuts S. A strongly connected digraph D is called λ′k-connected when λ ′ k(D) exists. In this paper we characterize all λ-connected tournaments, multipartite tournaments, local tournaments and in-tournaments. In addition, we determine the λ′2-connected tournaments and local tournaments. 1 Terminology and preliminary results We consider finite digraphs without loops, multiple arcs and directed cycles of length two. For any digraph D the vertex set is denoted by V (D) and the arc set by E(D). We define the order ofD by n = n(D) = |V (D)| and the size bym = m(D) = |E(D)|. If uv is an arc of a digraph D, then v is a positive neighbor of u and u a negative neighbor of v, and we also say that u dominates v. If A and B are two disjoint subdigraphs of D such that every vertex of A dominates every vertex of B, then we say A dominates B, denoted by A → B. The outset N(u) = N D (u) and the inset N(u) = N− D (u) of a vertex u is the set of positive neighbors and negative neighbors ∗ [email protected] 270 DIRK MEIERLING, LUTZ VOLKMANN AND STEFAN WINZEN of u, respectively. The numbers d(u) = d+D(u) = |N (u)| and d(u) = d−D(u) = |N(u)| are the out-degree and the in-degree of the vertex u. By a cycle of a digraph we mean a directed cycle. A cycle of length p is also called a p-cycle. A digraph D is vertex pancyclic if every vertex of D is contained in a p-cycle for all p between 3 and |V (D)|. IfD is a digraph andX ⊆ V (D), thenD[X] is the subdigraph induced byX. Two vertices u and v of a digraph are adjacent if u → v or v → u. Two vertex-disjoint subdigraphs A and B of a digraph D are complementary, if V (D) = V (A) ∪ V (B). A digraph is called cycle complementary, if it has two complementary cycles. If C = x1x2 . . . xnx1 is a cycle, then the second power of the cycle C consists of C and the arcs xixi+2 for i = 1, 2, . . . , n where xn+j = xj for j = 1, 2. If we replace every arc uv by vu in a digraph D, then we call the resulting digraph the converse of D. A digraph D is strongly connected or simply strong if for every pair u, v of vertices there exists a directed path from u to v in D. A digraph D with at least k+1 vertices is k-connected if for every set A of at most k − 1 vertices, the subdigraph D − A is strong. The connectivity of a digraph D, denoted by κ(D), is then defined to be the largest value k such that D is k-connected. A digraph D is k-arc-connected if for any set S of at most k−1 arcs the subdigraph D−S is strong. The arc-connectivity λ(D) of a digraph D is defined as the largest value of k such that D is k-arc-connected. A c-partite or multipartite tournament is an orientation of a complete c-partite graph. A tournament is a c-partite tournament with exactly c vertices. A digraphD is a local tournament, if for every vertex u the out-neighborhood as well as the in-neighborhood of u induce tournaments. A digraph D is an in-tournament, if for every vertex u the in-neighborhood of u induces a tournament. For other graph theory terminology we follow Bang-Jensen and Gutin [2]. For strongly connected digraphs D, Volkmann [12] defined the following kinds of restricted arc-connectivity. An arc set S of D is a restricted arc-cut of D if D − S has a non-trivial strong component D1 such that D−V (D1) contains an arc. The restricted arc-connectivity λ(D) is the minimum cardinality over all restricted arc-cuts S. A strongly connected digraph D is called λ-connected, if λ(D) exists. Let k ≥ 2 be an integer. An arc set S of D is a k-restricted arc-cut of D if D − S contains at least k non-trivial strong components. The k-restricted arc-connectivity λ′k(D) is the minimum cardinality over all k-restricted arc-cuts S. A strongly connected digraph D is called λ′k-connected, if λ ′ k(D) exists. Proposition 1.1 (Volkmann [12] 2007). Let k ≥ 2 be an integer. A strongly connected digraph D is λ′k-connected, if and only if D contains at least k pairwise vertex-disjoint cycles. Observation 1.2. It is well-known (cf. Bang-Jensen and Gutin [2], p. 554) that the problem of finding at least k ≥ 2 vertex-disjoint cycles in a digraph is NP-complete. Applying Proposition 1.1, we observe that the recognition problem, whether λ′k(D) exists for a strongly connected digraph D, is NP-complete too. ARC-CONNECTIVITY OF GENERALIZED TOURNAMENTS 271 In this paper we will characterize the λ′2-connected local tournaments and tournaments. These characterizations (cf. Theorem 3.1 and Corollary 3.2) show that the recognition problem, whether a strongly connected local tournament or tournament of order n and sizem is λ′2-connected, is solvable in timeO(n(n+m)) (cf. Remark 4.1). In addition, we characterize all λ-connected tournaments, multipartite tournaments, local tournaments and in-tournaments. The following results play an important role in our investigations. Theorem 1.3 (Moon [9] 1966). Every strong tournament is vertex pancyclic. Theorem 1.4 (Bondy [4] 1976). Each strong c-partite tournament contains an m-cycle for each m ∈ {3, 4, . . . , c}. Let TR be the 3-regular tournament of order seven consisting of the cycle x1 x2 x3 x4 x5 x6 x7 x1 such that x1 → x3 → x5 →x1 → x6→ x2→ x7 → x3 → x6→ x4 → x2→ x5 → x7 → x4 → x1. Notice that TR is the unique Hadamard tournament of order 7 which contains no transitive subtournament of order 4. Theorem 1.5 (Reid [10] 1985). Let T be a 2-connected tournament of order n ≥ 6. If T 6= TR, then T contains two vertex-disjoint cycles of lengths 3 and n− 3. Theorem 1.6 (Bang-Jensen [1] 1990). Let D be a strongly connected local tournament, and let S be a minimal separating set of D. The strong components of D−S are tournaments and they can be ordered in a unique way D1, D2, . . . , Dp such that there are no arcs from Dj to Di for j > i, and Di → Di+1 for i = 1, 2, . . . , p− 1. Theorem 1.7 (Bang-Jensen, Huang, Prisner [3] 1993). An in-tournament is Hamiltonian if and only if it is strong. Theorem 1.8 (Bang-Jensen, Huang, Prisner [3] 1993). Let D be a strong in-tournament, and let S be a minimal separating set of D. The strong components of D − S can be ordered in a unique way D1, D2, . . . , Dp such that there are no arcs from Dj to Di for j > i, and there exists a vertex xi ∈ V (Di) such that xi → Di+1 for i = 1, 2, . . . , p− 1. Theorem 1.9 (Guo, Volkmann [5] 1994). Every partite set of a strongly connected c-partite tournament D contains at least one vertex that lies on cycles of each length m for m ∈ {3, 4, . . . , c}. Let D GV be the local tournament of order 6 consisting of the cycle u1u2u3u4u5u6u1 such that u1 → u3 → u6 → u2 → u4 → u6 and u2 → u5 → u3. 272 DIRK MEIERLING, LUTZ VOLKMANN AND STEFAN WINZEN Let D GV be the local tournament of order 7 consisting of the cycle v1v2v3v4v5v6v7v1 such that v3 → v5 → v7 → v2 → v5, v6 → v1 → v3 → v6 → v4 → v2 and v1 → v4 → v7. Theorem 1.10 (Guo, Volkmann [6], [7] 1994, 1996). Let D be a 2-connected local tournament of order n ≥ 6. Then D is cycle complementary, if and only if D 6= TR, D 1 GV , D 2 GV and D is not the second power of an odd cycle.