Pierre Ille

Absorbing sets in arc-coloured tournaments

Let T be a tournament whose arcs are coloured with k colours. Call a subset X of the vertices of T absorbing if from each vertex of T not in X there is a monochromatic directed path to some vertex in X. We consider the question of the minimum size of absorbing sets, extending known results and using new approaches. The greater part of the paper deals with finite tournaments, the last section treats infinite ones. In each case questions are suggested, both old and new.

Minimal indecomposable graphs

Abstract Let G = (V, E) be a graph, a subset X of V is an interval of G whenever for a, b ∈ X and x ∈ V − X, (a, x) ∈ E (resp. (x, a) ∈ E) if and only if (b, x) ∈ E (resp. (x, b) ∈ E). For instance, 0, x, where x ∈ V, and V are intervals of G, called trivial intervals. A graph G is then said to be indecomposable when all of its intervals are trivial. In the opposite case, we will say that G is decomposable. We now introduce the minimal indecomposable graphs in the following way. Given an indecomposable graph G = (V, E) and vertices x1, …, xk of G, G is said to be minimal for x1, …, xk whenever for every proper subset W of V, if x1, …, xk ∈ W and if |W| ⩾ 3, then the induced subgraph G(W) of G is decomposable. In this paper, we characterize the minimal indecomposable graphs for one or two vertices and we describe in a more precise manner the minimal indecomposable symmetric graphs, posets and tournaments.

Indecomposability and duality of tournaments

Abstract Let T =( V , A ) be a tournament. A subset X of V is an interval of T provided that for a , b ∈ X and for x ∈ V − X , ( a , x )∈ A if and only if ( b , x )∈ A . For example, ∅,{ x }, where x ∈ V , and V are intervals of T, called trivial intervals. A tournament is said to be indecomposable if all of its intervals are trivial. In another respect, with each tournament T =( V , A ) is associated the dual tournament T ★ =( V , A ★ ) defined as: for x , y ∈ V ,( x , y )∈ A ★ if ( y , x )∈ A . A tournament T is said to be self-dual if T and T ★ are isomorphic. The paper characterizes the finite tournaments T =( V , A ) fulfilling: for every proper subset X of V, if the subtournament T ( X ) of T is indecomposable, then T ( X ) is self-dual. The corollary obtained is: given a finite and indecomposable tournament T =( V , A ), if T is not self-dual, then there is a subset X of V such that 6⩽| X |⩽10 and such that T ( X ) is indecomposable without being self-dual. An analogous examination is made in the case of infinite tournaments. The paper concludes with an introduction of a new mode of reconstruction of tournaments from their proper and indecomposable subtournaments.

Reconstruction of Posets with the Same Comparability Graph

Given two finite posetsPandP? with the same comparability graph, we show that if |V(P)|?4 and if for allx?V(P),P?x?P??x, thenP?P?. This result leads us to characterize the finite posetsPsuch that for allx?V(P),P?x?P*?x.

Note: A proof of a conjecture of Sabidussi on graphs idempotent under the lexicographic product

In 1960, Sabidussi conjectured that if a graph G is isomorphic to the lexicographic product G[G], then the wreath product of Aut(G) by itself is a proper subgroup of Aut(G[G]). A positive answer is provided by constructing an automorphism @J of G[G] which satisfies: for every vertex x of G, there is an infinite subset I(x) of V(G) such that @J({x}xV(G))=I(x)xV(G).

Cut-primitive directed graphs versus clan-primitive directed graphs

Abstract Given a directed graph G = (V,A), a subset X of V is a clan of G provided that for a, b ∈ X and x ∈ V \ X, (a, x) ∈ A if and only if (b, x) ∈ A, and similarly for (x, a) and (x, b). For instance, ∅, V and {x}, where x ∈ V, are clans of G, called trivial. A directed graph is clan-primitive if all its clans are trivial. Given a directed graph G = (V,A), a subset X of V is a cut of G if X and V \ X are clans of G. For example, ∅ and V are cuts of G, called trivial. A directed graph is cut-primitive if all its cuts are trivial. Ehrenfeucht and Rozenberg [Theoret. Comput. Sci. 70: 343–358, 1990] proved: Given a clan-primitive directed graph G = (V,A), if X is a subset of V such that |X| ≥ 3, |V \ X| ≥ 2 and G[X] is clan-primitive, then there are x ≠ y ∈ V \ X such that G[X ∪ {x, y}] is clan-primitive. We show: Given a clan-primitive directed graph G = (V,A), if X is a proper subset of V such that |X| ≥ 3 and G[X] is cut-primitive, then there are x, y ∈ V \ X such that G[X ∪ {x, y}] is cut-primitive and {x}, {y} are maximal proper clans of G[X ∪ {x, y}].

Cyclic Extensions of Order Varieties

We study complexity problems involving three sorts of combinational structures: cyclic orders, order varieties and cycles. It is known that the problem of telling whether a cyclic order is included in some cycle is NP-complete. We show that the same question for order varieties, a special class of cyclic orders, is in L. For this, we relate the entropy relation between partial orders to the forcing relation of graph theory.

The {-2,-1}-selfdual and decomposable tournaments

Abstract We only consider finite tournaments. The dual of a tournament is obtained by reversing all the arcs. A tournament is selfdual if it is isomorphic to its dual. Given a tournament T, a subset X of V (T) is a module of T if each vertex outside X dominates all the elements of X or is dominated by all the elements of X. A tournament T is decomposable if it admits a module X such that 1 < |X| < |V (T)|. We characterize the decomposable tournaments whose subtournaments obtained by removing one or two vertices are selfdual. We deduce the following result. Let T be a non decomposable tournament. If the subtournaments of T obtained by removing two or three vertices are selfdual, then the subtournaments of T obtained by removing a single vertex are not decomposable. Lastly, we provide two applications to tournaments reconstruction.

Criticality of switching classes of reversible 2-structures labeled by an Abelian group

Abstract Let V be a finite vertex set and let (𝔸, +) be a finite abelian group. An 𝔸-labeled and reversible 2-structure defined on V is a function g : (V × V) \ {(v, v) : v ∈ V } → 𝔸 such that for distinct u, v ∈ V, g(u, v) = −g(v, u). The set of 𝔸-labeled and reversible 2-structures defined on V is denoted by ℒ(V, 𝔸). Given g ∈ ℒ(V, 𝔸), a subset X of V is a clan of g if for any x, y ∈ X and v ∈ V \ X, g(x, v) = g(y, v). For example, ∅, V and {v} (for v ∈ V) are clans of g, called trivial. An element g of ℒ(V, 𝔸) is primitive if |V | ≥ 3 and all the clans of g are trivial. The set of the functions from V to 𝔸 is denoted by 𝒮(V, 𝔸). Given g ∈ ℒ(V, 𝔸), with each s ∈ 𝒮(V, 𝔸) is associated the switch gs of g by s defined as follows: given distinct x, y ∈ V, gs(x, y) = s(x) + g(x, y) − s(y). The switching class of g is {gs : s ∈ 𝒮(V, 𝔸)}. Given a switching class 𝔖 ⊆ ℒ(V, 𝔸) and X ⊆ V, {g↾(X × X)\{(x,x):x∈X} : g ∈ 𝔖} is a switching class, denoted by 𝔖[X]. Given a switching class 𝔖 ⊆ ℒ(V, 𝔸), a subset X of V is a clan of 𝔖 if X is a clan of some g ∈ 𝔖. For instance, every X ⊆ V such that min(|X|, |V \ X|) ≤ 1 is a clan of 𝔖, called trivial. A switching class 𝔖 ⊆ ℒ(V, 𝔸) is primitive if |V | ≥ 4 and all the clans of 𝔖 are trivial. Given a primitive switching class 𝔖 ⊆ ℒ(V, 𝔸), 𝔖 is critical if for each v ∈ V, 𝔖 − v is not primitive. First, we translate the main results on the primitivity of 𝔸-labeled and reversible 2-structures in terms of switching classes. For instance, we prove the following. For a primitive switching class 𝔖 ⊆ ℒ(V, 𝔸) such that |V | ≥ 8, there exist u, v ∈ V such that u ≠ v and 𝔖[V \ {u, v}] is primitive. Second, we characterize the critical switching classes by using some of the critical digraphs described in [Y. Boudabous and P. Ille, Indecomposability graph and critical vertices of an indecomposable graph, Discrete Math. 309 (2009) 2839–2846].

Decomposition tree and indecomposable coverings

Let G = (V,A) be a directed graph. With any subset X of V is associated the directed subgraph G[X ] = (X,A ∩ (X × X)) of G induced by X . A subset X of V is an interval of G provided that for a, b ∈ X and x ∈ V \ X , (a, x) ∈ A if and only if (b, x) ∈ A, This work was supported, in part, by an NSF EPSCoR grant EPS-0346476 and by a Nebraska Research Initiative (NRI) grant on high performance wireless networks. This research was done while the third author was visiting the University of Nebraska – Lincoln. 38 A. Breiner, J. Deogun and P. Ille and similarly for (x, a) and (x, b). For example ∅, V, and {x}, where x ∈ V , are intervals of G which are the trivial intervals. A directed graph is indecomposable if all its intervals are trivial. Given an integer k > 0, a directed graph G = (V,A) is called an indecomposable kcovering provided that for every subset X of V with |X | ≤ k, there exists a subset Y of V such that X ⊆ Y , G[Y ] is indecomposable with |Y | ≥ 3. In this paper, the indecomposable k-covering directed graphs are characterized for any k > 0.