Marthe Bonamy, Paul Dorbec, Paul Ouvrard. July 6, University of Bordeaux

Transcription

1 RECONFIGURING DOMINATING SETS UNDER TOKEN SLIDING Marthe onamy, Paul Dorbec, Paul Ouvrard July 6, 2017 University of ordeaux

2 Domination Definition A dominating set in a graph G (V, E) is a subset D V such that every vertex not in D is adjacent to at least one member of D. 1/19

3 Graph reconfiguration Rule : each intermediate solution must be a dominating set, we are only allowed to slide a token along an edge! TS? 2/19

4 Graph reconfiguration TS? 2/19

5 Graph reconfiguration TS? 2/19

6 Graph reconfiguration TS? 2/19

7 Graph reconfiguration TS? 2/19

8 Example not always possible! TS? 3/19

9 General definitions Dominating Set Reconfiguration Input : a graph G, two dominating sets A and of G We want : to transform step-by-step A into ; each intermediate solution must be a dominating set. Elementary operations token addition and removal (TAR) token jumping (TJ) token sliding (TS) 4/19

10 Equivalence TAR/TJ Notations Let G be a graph and A, be two dominating sets of size k of G. A TAR : we can reconfigure A into with the TAR model and each solution is of size at most k + 1. A T J : we can reconfigure A into with the TJ model. 5/19

11 Equivalence TAR/TJ Lemma Let G be a graph and A and be two dominating sets of G of size k. We have A TAR iff A T J. Proof : adapted from 1 (Theorem 1) T J TAR : easy. Replace each move u v by : first add v remove u each solution has size at most k + 1 We double the length of the sequence 1M. Kaminski, P. Medvedev, and M. Milanic. Complexity of independent set reconfigurability problems. Theoretical Computer Science, 439 :9 15, /19

12 Equivalence TAR/TJ TAR T J TAR-sequence of length 2n with configurations of size k or k + 1 Alternation of addition of v followed by a removal of u Replace by u T J v get a TJ-sequence of length n TAR-sequence of length 2n with configurations of any size There exists a subsequence which consists in the removal of a vertex u followed by the addition of a vertex v If u v remove the subsequence Otherwise, switch the order (Possibly reiterate the process) 7/19

13 Reduction Theorem [Haddadan et al.] Reconfiguring dominating sets under TAR(k + 1) is PSPACEcomplete. The problem is in PSPACE2 (Theorem 1). Lemma Reconfiguring dominating sets under TS(k) is PSPACE-complete in split graphs. 1Ito, T., Demaine, E.D., Harvey, N.J.A., Papadimitriou, C.H., Sideri, M., Uehara, R., Uno, Y.: On the complexity of reconfiguration problems. Theoretical Computer Science 412, pp (2011) 8/19

14 Complexity proof Proof : reduction from TJ in some graph G TS in the corresponding split graph. v 1 v 1 w 1 v 2 v 2 w 2 v 3 v 3 v 4 w 3 v 4 w 4 (a) G (b) G Figure: Graph G and the corresponding split graph G 9/19

15 Definitions G (V, E) with V {v 1, v 2,, v n } G i G[{v i, v i+1,, v n }] v maximum neighbor of u : v N[u] w N[u] : N[w] N[v] v is adjacent to all the vertices at distance 2 from u maximum neighborhood ordering (mno) : ordering on the vertices such that v i has a maximum neighbor in G i 10/19

16 Définitions A dually chordal graph is a graph which has a maximum neighborhood ordering. Lemma [Dorbec, Kosmrlj, Renault] Let G be a dually chordal graph. There exists a maximum neighborhood ordering v 1, v 2,, v n of G such that if the only maximum neighbor of v i in G i is itself, then v i is an isolated vertex in G i. We call such a mno a proper mno. 11/19

17 Example of dually chordal graph v 8 v 8 v 8 v 7 v 6 v 5 v 3 v 8 v 6 v 5 v 4 v 2 v 1 v 6 v 5 v 3 12/19

18 Minimum dominating sets reconfiguration Let A and be two minimum dominating sets of a dually chordal graph G. We want to reconfigure A into. Proof (sketch) : We distinguish a minimum dominating set canonical dominating set denoted C We show that both A and reconfigure into C Reversible operation one can reconfigure A into 13/19

19 Algorithm to compute the canonical dominating set Input : a dually chordal graph G, a proper mno Output : a canonical dominating set of G 1. Label all vertices with ounded 2. For i from 1 to γ(g) If v i is ounded Mark vertex v i Label mn(v i ) with Required For each vertex v N(mn(v i )) which is not Required, label it Free The vertices labeled Required form a dominating set. 14/19

20 Example v 8 v 7 v 6 v 5 v 3 v 4 v 2 v 1 15/19

21 Example v 8 v 7 v 6 v 5 v 3 v 4 v 2 v 1 15/19

22 Example v 8 v 7 v 6 v 5 v 3 v 4 v 2 v 1 15/19

23 Example v 8 v 7 v 6 v 5 v 3 R v 4 v 2 v 1 15/19

24 Example v 8 v 7 v 6 v 5 v 3 R F v 4 v 2 F v 1 F 15/19

25 Example v 8 v 7 v 6 v 5 v 3 R F v 4 v 2 F v 1 F 15/19

26 Example v 8 v 7 v 6 R v 5 v 3 R F v 4 v 2 F v 1 F 15/19

27 Example F v 8 F v 7 v 6 R v 5 v 3 R F v 4 F v 2 F v 1 F 15/19

28 Reconfiguration algorithm Let C {c 1, c 2,, c γ(g) } be the canonical dominating set Let m 1, m 2,, m γ(g) be the marked vertices (by the previous algorithm) Let D be the minimum dominating set that we want reconfigure into C Le x i be a vertex in N[m i ] D Proof (sketch) : i = 1 : w N[m 1 ], N[w] N[c 1 ] In particular : N[x 1 ] N[c 1 ] (D \ {x 1 }) {c 1 } is a dominating set of G (x 1, c 1 ) E(G) 16/19

29 Reconfiguration algorithm i i + 1 Suppose it is true at rank i i.e. (D \ {x 1,, x i }) {c 1,, c i } is a dominating set of G We denote v j m i+1 and v k x i+1 We distinguish two cases : v k v j : all the vertices before v j (in the mno) are already dominated N j [x i+1 ] N j [c i+1 ] (D \ {x 1,, x i, x i+1 }) {c 1,, c i, c i+1 } is also a dominating set of G (x i+1, c i+1 ) E(G) 17/19

30 Reconfiguration algorithm i i + 1 v k < v j : v k {c 1, c 2,, c i } (because v j is labeled ounded) all the vertices before v j (still in the mno) are already dominated If (v k, c i+1 ) E(G) : nothing to do Otherwise, the situation is the following : mn(v k ) v k v j ci+1 We can slide v k to c i+1 in two steps (via mn(vk)). 18/19

31 Conclusion Problem PSPACE-complete in some graph classes... Split graphs Planar graphs ounded treewidth graphs... but polynomial in other ones Interval graphs (for any k γ(g)) Dually chordal (for k γ(g)) Cographs Find a class for which computing a minimum dominating set is hard but for which reconfiguration can be done in polynomial time. 19/19

32 Conclusion Problem PSPACE-complete in some graph classes... Split graphs Planar graphs ounded treewidth graphs... but polynomial in other ones Interval graphs (for any k γ(g)) Dually chordal (for k γ(g)) Cographs Find a class for which computing a minimum dominating set is hard but for which reconfiguration can be done in polynomial time. Thank you for your attention! 19/19

33 TAR(k) TAR(k + 1) 20/19