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.
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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
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