A Richardson’s theorem version for Π -kernels

Discrete Mathematics 340 (2017) 984–987

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A Richardson’s theorem version for Π -kernels✩ Hortensia Galeana-Sánchez, Juan José Montellano-Ballesteros * Instituto de Matemáticas, UNAM, Mexico

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Article history: Received 27 October 2015 Received in revised form 20 September 2016 Accepted 9 January 2017 Available online 8 February 2017 Keywords: Richardson’s theorem Π -kernel

a b s t r a c t Let D = (V (D), A(D)) be a digraph, DP(D) be the set of directed paths of D and let Π be a subset of DP(D). A subset S ⊆ V (D) will be called Π -independent if for any pair {x, y} ⊆ S, there is no xy-path nor yx-path in Π ; and will be called Π -absorbing if for every x ∈ V (D) \ S there is y ∈ S such that there is an xy-path in Π . A set S ⊆ V (D) will be called a Π -kernel if S is Π -independent and Π -absorbing. Given x, y ∈ V (D) and Q ⊆ Π we will use (x→Q y) (respectively (x̸ →Q y)) to denote the fact that there is an (respectively, no) xy-path in Q. A subset Q ⊆ Π will be called transitive if whenever (x→Q y) and (y→Q z), then (x→Q z). Given a partition P = {Πi }i∈I of Π , the color digraph of D and I is the digraph CD (P ) with I as vertex-set and given any pair {j, k} ⊆ I , the arc jk belongs to CD (P ) if and only if there are x, y, w ∈ V (D) such that (x→Πj y) and (y→Πk w ). In this paper we prove the following result: Let D be a digraph and Π ⊆ DP(D). If there is a partition P = {Πi }i∈I of Π such that: For each i ∈ I , Πi is transitive; and CD (P ) has no odd cycles of order greater than 1, then D has Π - kernel. Some interesting previous results are obtained as a direct consequence of this result. © 2017 Elsevier B.V. All rights reserved.

1. Introduction For general concepts we may refer the reader to [1,2]. Let D = (V (D), A(D)) be a digraph. A set K ⊆ V (D) is said to be a kernel if it is both independent (a vertex in K has no successor in K ) and absorbing (a vertex not in K has a successor in K ). This concept was first introduced in [8] by Von Neumann and Morgenstern in the context of Game Theory as a solution for cooperative n-player games. In [4] Chvátal showed that deciding if a graph possesses a kernel is an NP-complete problem, and in [6] Fraenkel showed that it remains NP-complete for planar directed graphs with indegrees and outdegrees less than or equal to 2, and degrees less than or equal to 3. The concept of kernel is important to the theory of digraphs because it arises naturally in applications such as Nim-type games, logic and facility location to name a few. Several authors have been investigating sufficient conditions for the existence of kernels in digraphs, for a comprehensive survey see for example [3,6]. A classical result on the theory of kernels is the following due to Richardson and proved in [9]. Theorem 1 (Richardson’s Theorem). A finite digraph without directed cycles of odd length has a kernel. In this paper we present a wide generalization of Richardson’s theorem. 2. Notation and previous results Let D = (V (D), A(D)) be a digraph. A directed path P = (x0 , x1 , . . . , xn ) of D will be called an x0 xn -path. Given S1 , S2 ⊆ V (D), an S1 S2 -path is an xy-path where x ∈ S1 and y ∈ S2 (if S1 = {x} we will write xS-path and Sx-path instead of {x}S-path and ✩ Research partially supported by PAPIIT-México project IN108715, PAPIIT-México project IN104915 and by CONACYT 201X project 219840. Corresponding author. E-mail addresses: [email protected] (H. Galeana-Sánchez), [email protected] (J.J. Montellano-Ballesteros).

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http://dx.doi.org/10.1016/j.disc.2017.01.008 0012-365X/© 2017 Elsevier B.V. All rights reserved.

H. Galeana-Sánchez, J.J. Montellano-Ballesteros / Discrete Mathematics 340 (2017) 984–987

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S {x}-path, respectively). Let DP(D) be the set of directed paths of D and let Π be a subset of DP(D). A subset S ⊆ V (D) will be called Π -independent if for any pair {x, y} ⊆ S, there is no xy-path nor yx-path in Π ; and will be called Π -absorbing if for every x ∈ V (D) \ S there is an xS-path in Π . A set S ⊆ V (D) will be called a Π -kernel if S is Π -independent and Π -absorbing. Clearly, if Π is the set of arcs of D, a Π -kernel is a kernel of D, and if Π is the set of monochromatic paths of an m-colored digraph D, a Π -kernel is a kernel by monochromatic paths in D. Given a pair {x, y} ⊆ V (D) and Q ⊆ Π , we will use (x→Q y) (respectively (x̸ →Q y)) to denote the fact that there is an (respectively, no) xy-path in Q. In an analogous way, given S1 , S2 ⊆ V (D), we will use (S1 →Q S2 ) (respectively (S1 ̸ →Q S2 )) to denote the fact that there is an (respectively, no) S1 S2 -path in Q. The Π -closure of D is the digraph ClΠ (D) whose vertex-set is V (D) and the arc xy belongs to ClΠ (D) if and only if (x→Π y). We will say that Π is complete if for every pair {x, y} ⊆ V (D), (x→Π y) or (y→Π x) (and maybe both). A subset Π1 ⊆ Π will be called transitive if whenever (x→Π1 y) and (y→Π1 z), then (x→Π1 z). Given a partition P = {Πi }i∈I of Π , the P -closure of D is the arc-colored multidigraph ClP (D) whose set of vertices is V (D), and the arc xy with ‘‘color’’ k ∈ I belongs to ClP (D) if and only if (x→Πk y) (observe that if P = {Π }, then ClP (D) = ClΠ (D)). A subdigraph H of ClP (D) will be called heterochromatic if no two arcs of H have the same color. Observe that D has a Π -kernel if and only if ClΠ (D) has a kernel if and only if ClP (D) has a kernel. Given a partition P = {Πi }i∈I of Π , the color digraph of D and I is the digraph CD (P ) with I as vertex-set and given any pair {j, k} ⊆ I , the arc jk belongs to CD (P ) if and only if there are x, y, w ∈ V (D) such that (x→Πj y) and (y→Πk w ). Let P = {Πi }i∈I be a partition of Π , and let ClP (D) = (VP , AP ) and CD (P ) = (VI , AI ) be the P -closure of D and the color digraph of D and I , respectively. Clearly, the set of arcs AP is a subset of the directed paths of ClP (D), and the arc coloring of ClP (D) induces the partition {Ai }i∈I of AP (for each i ∈ I , Ai = {xy ∈ AP : xy has color i}). It is not hard to see that the color digraph of ClP (D) and I is isomorphic to CD (P ). ⋃ ′ Given VI′ ⊆ VI a subset of vertices of CD (P ), let Π ′ ⊆ DP(D) defined as Π ′ = ′ Πi and let P = {Πi }i∈V ′ be the i∈VI I partition of Π ′ . Thus, the P ′ -closure of D is the colored digraph ClP ′ (D) with V (D) as vertex-set and the arc xy with color j belongs to ClP ′ (D) if and only if there is j ∈ VI′ such that (x→Πj y). Observe that ClP ′ (D) is an spanning subdigraph of ClP (D). On the other hand, given VP∗ ⊆ VP = V (D) a subset of vertices of ClP (D), let ClP (D)[VP∗ ] = (VP∗ , A∗P ) be the subdigraph of ClP (D) induced by VP∗ , and let I ∗ ⊆ I be the set of colors that appear in ClP (D)[VP∗ ]. Once again, A∗P is a subset of the set of directed paths of ClP (D)[VP∗ ], and {A∗i }i∈I ∗ , with A∗i = {xy ∈ A∗P : xy has color i ∈ I ∗ }, is a partition of A∗P . It is not hard to see that the color digraph of ClP ∗ (D)[V ∗ ] and I ∗ is a subdigraph of CD (P )[I ∗ ] (the subdigraph of CD (P ) induced by I ∗ ⊆ VI ). In order to obtain the main result of this paper we will also need the following Theorem 2 ([7]). Let D be a digraph and Π ⊆ DP(D). If there is a partition {Π1 , Π2 } of Π such that Π1 and Π2 are both transitive, then D has a Π -kernel. 3. The main result Theorem 3. Let D = (V (D), A(D)) be a digraph and Π ⊆ DP(D). If there is a partition P = {Πi }i∈I of Π such that: (i) For each i ∈ I , Πi is transitive; (ii) CD (P ) has no odd cycles of order greater than 1; then D has Π - kernel. Proof. Let D = (V (D), A(D)) be a digraph and Π ⊆ DP(D). Let P = {Πi }i∈I be a partition of Π such that for each i ∈ I , Πi is transitive; and such that the color digraph CD (P ) = (VI , AI ) has no odd cycles of order greater than 1. We will proceed by induction on the cardinality of I . If |I | = 2 it follows that P = {Π1 , Π2 } is a partition of Π such that Π1 and Π2 are both transitive, which by Theorem 2 implies that D has a Π -kernel. Let suppose |I | ≥ 3. If CD (P ) is strongly connected, it follows ⋃ that CD (P ) is bipartite. Let {A, B} be the partition of VI and ⋃ let {A∗ , B∗ } be the partition of Π where A∗ = i∈A Πi and B∗ = i∈B Πi . We will see that A∗ and B∗ are both transitive. Let x, y, w ∈ V (D) such that (x→A∗ y) and (y→A∗ w ). Thus, for some i ∈ A∗ , (x→Πi y), and for some j ∈ A∗ , (y→Πj w ). If i ̸ = j, since {i, j} ⊆ A∗ , ij ̸∈ AI and therefore, by definition of CD (P ), there is no x, y, w ∈ V (D) such that (x→Πi y) and (y→Πj w). Thus, i = j and since Πi is transitive, (x→Πi w ) and therefore (x→A∗ w ) which implies that A∗ is transitive. In an analogous way we see that B∗ is transitive, and therefore P ∗ = {A∗ , B∗ } is a partition of Π such that A∗ and B∗ are both transitive. By Theorem 2, D has a Π -kernel. If CD (P ) = (VI , AI ) is not strongly connected, let⋃ G0 = (VI0 , A0I ) be a strongly connected terminal component of CD (P ), ⋃ 1 0 1 0 and let VI = VI \ VI . Let Π = i∈V 1 Πi and Π = i∈V 0 Πi . Observe that since CD (P ) is not strongly connected, VI1 ̸ = ∅. I

I

Let P 0 = {Πi }i∈V 0 be the partition of Π 0 , ClP 0 (D) = (VP 0 , AP 0 ) be the P 0 -closure of D and let {Aj }j∈V 0 , with Aj = {xy ∈ I

I

AP 0 : xy has color j ∈ VI0 }, be the partition of AP 0 . As we see in the previous section, the color digraph of ClP 0 (D) and VI0 , denoted as CCl 0 (D) (VI0 ), is a spanning subdigraph of CD (P ), and therefore CCl 0 (D) (VI0 ) has no odd cycles of order greater than 1. P P Since VI1 ̸ = ∅ it follows that |VI0 | < |I | which by induction hypothesis implies that ClP 0 (D) possesses a Π 0 -kernel. Let N0 be a Π 0 -kernel in ClP 0 (D). Observe that since Π 0 = AP 0 , N0 is a kernel of ClP 0 (D). Moreover, since VI0 ̸ = ∅, then AP 0 ̸ = ∅ and VP 0 \ N0 ̸ = ∅.

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By definition of ClP 0 (D), xy ∈ AP 0 if and only if there is j ∈ VI0 such that (x→Πj y). Thus, since VP 0 = V (D) it follows that N0 is Π -absorbent in D. Let us suppose N0 is not Π -independent in D and let u, v ∈ N0 such that for some j ∈ I , (u→Πj v ). Since N0 is a Π 0 -kernel, it follows that j ∈ VI1 . Claim 1. For every w ∈ VP 0 \ N0 and every i ∈ VI0 , (w̸ →Πi u). If for some w ∈ VP 0 \ N0 and some i ∈ VI0 we have that (w→Πi u), since (u→Πj v ) this implies that ij ∈ AI but i is a vertex of G0 which is a terminal component of CD (P ) and j ∈ VI1 = VI \ VI0 , and this is a contradiction. Let T = {z ∈ N0 : there is k ∈ N0 \ {z } such that (z →Πj k) for some j ∈ VI1 }. Observe that since N0 is not Π -independent in D, T ̸ = ∅. Let N1 = N0 \ T . Claim 2. N1 ̸ = ∅ Let suppose N1 = ∅. Since VP 0 \ N0 ̸ = ∅ and N0 is a Π 0 -kernel of ClP 0 (D), there is h ∈ VP 0 \ N0 and w ∈ N0 such that (h→Πi w ) with i ∈ VI0 which contradicts Claim 1. Claim 3. N1 is Π -independent in D. Since N1 ⊆ N0 it follows that N1 is Π 0 -independent, and by definition of T , there is no j ∈ VI1 and u, v ∈ N1 such that (u→Πj v ). Thus the claim follows. Claim 4. Given w ∈ V (D) \ N0 there is h ∈ N1 such that (w→Πj h) for some j ∈ VI0 . Let w ∈ V (D) \ N0 . Since N0 is Π 0 -absorbent in ClP 0 (D) and VP 0 = V (D), there is j ∈ VI0 and h ∈ N0 such that (w→Πj h). By Claim 1 and by definition of T , h ̸ ∈ T and the claim follows. Observe that if for every t ∈ T , (t →Π N1 ), then N1 is a Π -kernel of D. Let T ∗ = {t ∈ T : (t ̸ →Π N1 )}. Let ClP (D)[T ∗ ] = (V ∗ , A∗ ) be the subdigraph of ClP (D) induced by T ∗ , let I ∗ be the colors that appear in ClP (D)[T ∗ ] and {A∗i }i∈I ∗ be the partition of A∗ , with A∗i = {xy ∈ A∗ : xy has color i ∈ I ∗ }. Since T ∗ ⊆ N0 and N0 is Π 0 -independent, it follows that I ∗ ⊆ VI1 and that the color digraph of ClP (D)[T ∗ ] and I ∗ , denoted as CClP (D)[T ∗ ] (I ∗ ), is a subdigraph of CD (P )[I ∗ ]. Thus, CClP (D)[T ∗ ] (I ∗ ) has no odd cycles of order greater than 1, and since |VI1 | < |I |, by induction hypothesis ClP (D)[T ∗ ] has a Π 1 -kernel N2 . Claim 5. N2 is Π -independent in D Let x, y ∈ N2 and i ∈ I such that (x→Πi y). Since N2 ⊆ T ∗ , xy ∈ A∗ and therefore, since N2 is a Π 1 -kernel, it follows that i ∈ VI0 but since T ∗ ⊆ N0 and N0 is Π 0 -independent this is a contradiction. Now we will show that N1 ∪ N2 is a Π -kernel in D. By Claims 3 and 5, N1 and N2 are Π -independent in D. By definition of N1 , (N1 ̸ →Π N2 ) and of T ∗ , (N2 ̸ →Π N1 ). Thus N1 ∪ N2 is Π -independent in D. ( by definition ) Let x ∈ V (D) \ N1 ∪ N2 . If x ∈ T ∗ \ N2 it follows, since N2 is a Π 1 -kernel of ClP (D)[T ∗ ], that (x→Π N2 ). If x ∈ T \ T ∗ , by

definition of T we see that (x→Π N1 ). If x ∈ N0 \ T then x ∈ N1 and by Claim 4, for every x ∈ V (D) \ N0 , (x→Π N1 ). From here, the result follows. For the case when D is a tournament we have the following Theorem 4. Let T = (V (T ), A(T )) be a tournament and Π ⊆ DP(D) be complete. If there is a partition P = {Πi }i∈I of Π such that: (i) For each i ∈ I , Πi is transitive; (ii) There is no heterochromatic directed triangle in the P -closure of T with less than two symmetric arcs; then T has Π - kernel. Proof. Let T = (V (T ), A(T )) be a tournament; Π ⊆ DP(D) be complete and P = {Πi }i∈I be a partition of Π such that for each i ∈ I , Πi is transitive; and such that there is no heterochromatic directed triangle in the P -closure of T with less than two symmetric arcs. We will see that every directed cycle in the P -closure of T has a symmetric arc, which implies (see [5]) that the P -closure of T has a kernel, which directly implies that T has a Π - kernel. For this, let ClP (T ) be the P -closure of T , colored with the set of colors I and let suppose there is a directed cycle of minimum length γ = (x1 , x2 , . . . , xr ) in ClP (T ) with non-symmetric arcs. Since for each i ∈ I , Πi is transitive and γ has non-symmetric arcs, it follows that in γ appears at least two arcs of different colors q, k ∈ I , and thus there is j, with 0 ≤ j ≤ r, such that xj−1 xj has color q and xj xj+1 has color k (the indexes are taken modulo r).

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Claim 6. xj+1 xj−1 is not an arc of ClP (T ) Suppose xj+1 xj−1 is an arc of ClP (T ). If xj+1 xj−1 has color q, since Πq is transitive and xj−1 xj has color q, it follows that xj+1 xj has color q and belongs to ClP (T ), which is a contradiction since γ has non-symmetric arcs. Analogously we see that the color of xj+1 xj−1 is not k. So the arcs {xj xj+1 , xj+1 xj−1 , xj−1 xj } induces a heterochromatic triangle C in ClP (T ), but since xj−1 xj and xj xj+1 are non-symmetric arcs, C has at most one symmetric arc, which is a contradiction. Since Π is complete, and by Claim 6, it follows there is an xj−1 xj+1 -path in Π , and therefore there is ℓ ∈ I such that (xj−1 →Πℓ xj+1 ). Hence xj−1 xj+1 is an arc of ClP (T ) and, again by Claim 6, is a non-symmetric arc of ClP (T ). Thus (x1 , x2 , . . . , xj−1 , xj+1 , . . . , xr ) is a directed cycle in ClP (T ) with non-symmetric arcs and with smaller length than γ , which is a contradiction. Given an arc-colored tournament T , with set of colors I , let Π ⊆ DP(T ) be the set of monochromatic paths in T and P = {Πi }i∈I be the partition of Π such that for each i ∈ I , Πi is the set of monochromatic paths of T of color i ∈ I . It follows that Π is complete; that for each i ∈ I , Πi is transitive; and that the P -closure of T is just the closure by monochromatic paths of T . Theorem 4 implies the following corollary, which responds partially in the positive direction to the following question posed by Sand, Sauer and Woodrow in [10]: Question. Does any finite 3-colored tournament without heterochromatic 3-cycles admit an absorbing vertex? Corollary 1. Let T be a 3-colored tournament. If there is no heterochromatic directed triangle in the closure by monochromatic paths of T with less than two symmetric arcs, then T has an absorbing vertex by monochromatic paths. As a consequence of Theorem 4 we have Theorem 5 (Richardson’s Theorem). A finite digraph without directed cycles of odd length has a kernel. Proof. Let D = (V (D), A(D)) be a finite digraph without directed cycles of odd length. Let Π = A(D) and P = {e}e∈A(D) be the partition of Π (each element of the partition is a single arc of D). It follows that each element of P is transitive, and that the color digraph of D and A(D) is the line digraph L(D) of D. Claim 7. If D has no directed cycles of odd length then L(D) has no directed cycles of odd length. Let C = (x1 , . . . , xn ) be a directed cycle of odd length in L(D) and let [n] = {1, . . . , n}. For each i ∈ [n], let xi = yi wi ∈ A(D). Since for each i ∈ [n], xi xi+1 is an arc in L(D) (taken the indexes modulo n) it follows that for each i ∈ [n], wi = yi+1 and therefore C induces a closed directed trail (w1 , . . . , wn ) in D of odd length which implies there is a directed cycle of odd length in D, and the claim follows. Since every element of Π is transitive and the color digraph of D and A(D) has no directed cycles of odd length, from Theorem 4 it follows that D has a Π -kernel, which, since Π = A(D), is a kernel of D. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

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