• Aucun résultat trouvé

locally equivalent

N/A
N/A
Protected

Academic year: 2022

Partager "locally equivalent"

Copied!
3
0
0

Texte intégral

(1)

ENUMERATION OF LOCAL COMPLEMENTATION ORBIT

Keywords: rank-width; rank-decomposition; pivot-minor; local complementation; locally equivalent; enumeration algorithm; vertex- minor

Internship atLimos, Clermont-Ferrand, supervised by Mamadou M. Kant´e.

Introduction. Rank-widthis a complexity measure introduced by Oum and Seymour [8, 7]. Rank-width is interesting for several reasons.

(1) It is equivalent to clique-width, a complexity measure introduced by Courcelle et al. [3], that generalises the well-known complexity measure tree-width introduced by Robertson and Seymour in their graph minors series.

(2) It is algorithmically more interesting than clique-width because we can recognise in polynomial time graphs of rank-width at most k (for fixed k)

(3) It shares with tree-width many structural properties (its is for in- stance related to the theory of matroids and is related to thevertex- minor relation [7]).

(4) . . .

Local complementation. The vertex set of a graphGis denoted byV(G), its edge set by E(G), and their sizes are denoted respectively by nand m.

Given a graph G and a vertexx of G, the local complementation ofG at x is the graph, denoted by G∗x, obtained from G by replacing G[N(x)] by its complement. Of course, G∗x∗x =G. A graph H is locally equivalent to a graph G ifH can be obtained from G by a sequence of local comple- mentations [2, 4, 5]. Observe that being locally equivalent is an equivalence relation. One can check in polynomial time whether two graphs on the same vertex set are locally equivalent in polynomial time [1, 5]. The local complementation operation is used to define the quasi-order vertex-minor on finite graphs and is related to the complexity measurerank-width [7]. A graph H is a vertex-minor of a graph G ifH is an induced subgraph of a graph locally equivalent to G. For instance circle graphs are characterised by a finite list of graphs to exclude as a vertex-minor [2], and graphs of bounded rank-width are characterised by a finite list of graphs to exclude as vertex-minors [7].

It is still open, for a fixed graph H, whether one can check in polyno- mial time if a given graph G containsH as a vertex-minor or not. In this internship we are interested in the listing of the orbit of a graph (wrt lo- cal complementation) in order to understand why checking vertex-minor is difficult.

1

(2)

2 ENUMERATION OF LOCAL COMPLEMENTATION ORBIT

Enumeration Algorithms. Ahypergraph His a pair (VH,EH) with EH⊆ 2V \ {∅}.

Let D be a family of subsets of the vertex set of a given hypergraph H on n vertices and m hyperedges. An enumeration algorithm for D lists the elements of D without repetitions. Anenumeration problem forD asks for an enumeration algorithm for D. The running time of an enumeration algorithm A is said to beoutput polynomial if there is a polynomialp(x, y) such that all the elements ofDare listed in time bounded byp((n+m),|D|).

Assume now thatD1, . . . , D` are the elements ofDenumerated in the order in which they are generated by A. Let us denote by T(A, i) the time A requires until it outputsDi, alsoT(A, `+1) is the time required byAuntil it stops. Letdelay(A,1) =T(A,1) anddelay(A, i) =T(A, i)−T(A, i−1). The delayofAis max{delay(A, i)}. AlgorithmAruns inincremental polynomial time if there is a polynomial p(x, i) such that delay(A, i) ≤ p(n+m, i).

Furthermore Ais apolynomial delay algorithm if there is a polynomialp(x) such that the delay of A is at most p(n+m). Finally A is a linear delay algorithm ifdelay(A,1) is bounded by a polynomial inn+manddelay(A, i) is bounded by a linear function in n+m.

The Goal : Enumeration of (Non) Locally Equivalent Graphs. We are interested in the following questions

Problem. Lc-Equivalent(resp. Iso-Lc-Equivalent) Input. A graphG.

Output. The set of (resp. non-isomorphic) graphs locally equivalent toG.

Problem. Gen-Lc-Equivalent(resp. Gen-Iso-Lc-Equivalent) Input. A positive integern.

Output. The set of (resp. non isomorphic) non locally equivalent graphs withnvertices.

If H is locally equivalent to G, then the distance between H and G is the minimum number of local complementations to apply toGto obtainH.

From [4] the distance between two graphs is bounded by max{n+ 1,109n}.

Therefore, Lc-Equivalent admits an output-polynomial time algorithm.

Similarly for Iso-Lc-Equivalent ifG belongs to a graph class closed un- der local complementation and admitting a polynomial time Isomorphism Testing algorithm. However, the algorithm uses exponential space and consists in doing the following: Generate all graphs locally equivalent to G at distance 1, then those at distance 2, and so on until the generation of graphs at distance max{n+ 1,109 n}; keep a graph (or one representative of each isomorphism class) only if it was not generated on previous steps.

We are in a first step interesting in programming this procedure (or equiv- alent ones) using SAGE, and have on hand a library for local complementa- tion, vertex-minor and the implementation of the algorithm by Bouchet or Fon-Der-Flaass. In a second step, we can start thinking about the following questions.

Question 1.: Does Lc-Equivalent admit a polynomial delay and polynomial space algorithm? Is it the case also forIso-Lc-Equivalent if we assume that the input is in a graph class closed under local

(3)

ENUMERATION OF LOCAL COMPLEMENTATION ORBIT 3

complementation and admitting a polynomial time Isomorphism Testing?

Question 2.: AreGen-Lc-EquivalentandGen-Iso-Lc-Equivalent doable with polynomial delay and polynomial space in graph classes closed under local complementation and admitting a polynomial timeIsomorphism Testingalgorithm?

All materials and needed explanations will be provided and for further information please feel free to contact me atkante@isima.fr. You can find the definition of rank-width and clique-width in my Phd Thesis, for more materials on enumeration my Habilitation, both available online [9]. I can provide electronic copies of the papers, in the references, if not available in the authors’ webpages.

References

[1] Andr´e Bouchet. Recognizing locally equivalent graphs.Discrete Mathematics 114(1- 3): 75-86 (1993).

[2] Andr´e Bouchet. Circle Graph Obstructions.J. Comb. Theory, Ser.B 60(1): 107-144 (1994).

ask others

[3] Bruno Courcelle, Stephan Olariu. Upper bounds to the clique width of graphs. Dis- crete Applied Mathematics 101(1-3): 77-114 (2000)

[4] D.G. Fon-Der-Flaass. Distance between locally equivalent graphs (in Russian).

Metody Diskret. Analiz. 48: 85-94 (1989).

[5] D.G. Fon-Der-Flaass. On the local complementations of simple and directed graphs.

In A.D. Korshunov (ed),Discrete Analysis and Operations Research15-34(1996) [6] Mamadou Kant´e, Vincent Limouzy, Arnaud Mary, Lhouari Nourine. On the Enu-

meration of Minimal Dominating Sets and Related Notions.SIAM J. Discrete Math.

28(4): 1916-1929 (2014).

[7] Sang-il Oum. Rank-width and vertex-minors. J. Comb. Theory, Ser. B 95(1): 79-100 (2005).

[8] Sang-il Oum, Paul D. Seymour. Approximating clique-width and branch-width. J.

Comb. Theory, Ser. B 96(4): 514-528 (2006)

[9] Mamadou M. Kant´e. Webpage:http://www.isima.fr/∼kante/research.php

Références

Documents relatifs

Lexical, Morphosyntactical, and Pragmatic Aspects Does a teacher at a distance produce the same kind of discourse as a “traditional” colleague, just because he does not see

The differences are that : (i) the cross section d a , or da,, is replaced by the real photon cross section ; (ii) in evaluating this cross section, the kinematics

In this article, exponential contraction in Wasserstein distance for heat semigroups of diffusion processes on Riemannian manifolds is established under curvature conditions where

Abstract—In this paper, the generation of 16-QAM and 64-QAM space-time trellis codes (STTCs) for several transmit antennas is considered.. The main problem with an exhaustive search

UTILISATION M is e à jo u r ❋✐❣✉r❡ ✶ ✕ ❙❝❤é♠❛ s②♥t❤ét✐q✉❡ ❞❡ ❧❛ ❣é♠♦❞é❧✐s❛t✐♦♥✱ ❝♦♠♣♦rt❛♥t ❧❡s ❞✐✛ér❡♥ts t②♣❡s ❞♦♥✲ ♥é❡s ✉t✐❧✐sé❡s✱

In Section 2, we introduce the mathematical model, we present briefly a formal derivation of equivalent conditions and we state uniform estimates for the solution of the

[16] ——, “Comments on CuBICA: Independent component analysis by simultaneous third- and fourth-order cumulant diagonalization,” IEEE Transactions on Signal Processing, vol.

/ La version de cette publication peut être l’une des suivantes : la version prépublication de l’auteur, la version acceptée du manuscrit ou la version de l’éditeur. Access