The generalized split probe problem 1

Simone Dantas

IME, Universidade Federal Fluminense, Brasil.

Luerbio Faria

IME, Universidade do Estado do Rio de Janeiro, Brasil.

Celina M. H. de Figueiredo

COPPE, Universidade Federal do Rio de Janeiro, Brasil.

Rafael B. Teixeira

ICE, Universidade Federal Rural do Rio de Janeiro, Brasil.

Abstract

A generalized split (k, l) partition is a vertex set partition into at most k indepen-dent sets and l cliques. We prove that the (2, 1) partitioned probe problem isin P whereas the (2, 2) partitioned probe is NP-complete. The full complex-ity dichotomy into polynomial time and NP-complete for the class of generalizedsplit partitioned probe problems establishes (2, 2) as the first NP-complete self-complementary partitioned probe problem, and answers negatively the PGC con-jecture by finding a polynomial time recognition problem whose partitioned probeversion is NP-complete.

Keywords: Probe graphs, partition problems, graph sandwich problems, graphalgorithms, computational complexity.

Available online at www.sciencedirect.com

Electronic Notes in Discrete Mathematics 44 (2013) 39–45

1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved.

www.elsevier.com/locate/endm

http://dx.doi.org/10.1016/j.endm.2013.10.007

1 Introduction

We say that a graph G = (V,E) is a sandwich graph for the pair G1 =(V,E1), G2 = (V,E2) if E1 ⊆ E ⊆ E2. The graph sandwich problem forproperty Π is defined by Golumbic et al. as follows [10]:

graph sandwich problem for property ΠInstance: Vertex set V , forced edge set E1, optional edge set E2 \ E1.Question: Is there a graph G = (V,E) such that E1 ⊆ E and E ⊆ E2,and G satisfies property Π?

Golumbic et al. [10] have considered sandwich problems with respect toseveral subclasses of perfect graphs, and proved that the graph sandwichproblem for split graphs (a two part sandwich problem) remains in P.The full complexity dichotomy into P or NP-complete of the three nonemptypart [13] and of the external constraint four nonempty part [14,15] sandwichproblems were subsequently obtained.

A generalized split (k, l) partition of a graph G is a partition of its vertexset into at most k independent sets and l cliques. A graph is (k, l) if it admitsa (k, l) partition. The full complexity dichotomy of (k, l) graph recognitionhas been determined. If k = 3 and l = 0 then the corresponding problem is3-coloring, which implies that the recognition of (k, l) graphs is NP-complete,whenever k ≥ 3 or l ≥ 3. For the remaining values of k and l, the problem ispolynomial: (1, 1) graphs are the split graphs; (2, 0) graphs are the bipartitegraphs; the polynomial-time recognition of (2, 1) graphs and consequently ofthe complementary class of (1, 2) graphs was established in [3]; the polynomialtime recognition of (2, 2) graphs was established in [3] and independently in [9].

The complexity dichotomy of generalized split graph recognition problemdescribed above implies the following complexity status for the graph sandwichproblem: for (k, l) graphs is NP-complete, whenever k ≥ 3 or l ≥ 3; andtogether with the hereditary properties defining (1, 0) and (2, 0) graphs, andthe ancestral properties defining (0, 1) and (0, 2) graphs, the respective graphsandwich problems are polynomial. The full dichotomy was obtained withtwo additional NP-completeness proofs: for (2, 1) graphs and consequentlyfor the complementary class of (1, 2), and for (2, 2) graphs [6]. These resultscompletely classify the complexity of the (k, l)-graph sandwich problem asfollows: the problem is NP-complete, if k + l > 2; and polynomial otherwise.

1 Partially supported by CNPq and FAPERJEmail: sdantas@im.uff.br, luerbio@cos.ufrj.br, celina@cos.ufrj.br,rafaelbt@ufrrj.br

S. Dantas et al. / Electronic Notes in Discrete Mathematics 44 (2013) 39–4540

The perfect graph sandwich is the only problem posed in [10] thatremains open, and rises a natural question: is the graph sandwich prob-lem for a self-complementary graph class C in P? The answer is positive forsplit [10], cographs [10] and P4-sparse [8] graphs; and negative for permuta-tion [10] and C5-free [7] graphs.

Polynomial sandwich problems are rare, so the particular instance wherethe optional edges of E2 \ E1 are required to have both endpoints in an in-dependent set of the forced graph G1 was considered as the partitionedprobe problem. Several partitioned probe problems have been studied, allof them so far classified as polynomial: cographs [12], P4-parse [4], permutationgraphs [5], threshold [1], chordal graphs [2], chain graphs [11], and triviallyperfect graphs [1], leading to the Probe Graph Conjecture (PGC): “Parti-tioned probe graphs of graph class C are polynomially recognizable wheneverC is polynomially recognizable” [12]. Most of the studied graph classes C areperfect and self-complementary leading to the PPGC conjecture: “perfectgraph partitioned probe problem is in P” [4].

The studies on the (k, l) partitioned probe problem were focused on thoseproblems which are interesting in terms of their complexity, i.e., NP-completesandwich problems and polynomial recognition problems: properties (2, 1), itscomplementary property (1, 2), and the self-complementary property (2, 2).

Analogously to recognition problems, the sandwich problem with respectto a graph class C has the same complexity (polynomial or NP-complete)as the sandwich problem with respect to C, the complementary graph classcontaining for each G ∈ C, the complement graph G [10]. On the other hand,there may exist a graph class C for which the partitioned probe problem is inP whereas the partitioned probe problem for C is NP-complete. We presentin this paper the first graph class for which the partitioned probe problem isNP-complete, and the graph class turns out to be self-complementary.

We establish the full complexity dichotomy for generalized split par-titioned probe problems, by proving that the problem is in P for both(2, 1) graphs and its complementary class (1, 2), and that the problem is NP-complete for the self-complementary class of (2, 2) graphs. Therefore, not allpartitioned probe problems are in P, contradicting de PGC conjecture [12], noteven when retricted to a self-complementary class, a contribution towards theperfect graph partitioned probe [4] and the perfect graph sand-wich [10] problems.

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2 Problems in P: (2, 1) and (1, 2) partitioned probe

We define the (2, 1) partitioned probe problem as follows:

(2, 1) partitioned probe problemInstance: Vertex set V , edge set E, partition (N,P ) of V , where N is anindependent set.Question: Is there a graph G′ = (V,E ′) such that E ⊆ E ′, all edges ofE ′ \ E have both endpoints in N , and G′ is a (2, 1) graph?

Algorithm 1 receives an instance of (2, 1) partitioned probe problemand returns a (2, 1) probe partition if there exists one. Algorithm 1 startsby testing for a necessary condition at Line 1 [3]. The polynomial aspect ofthis algorithm is based on the polynomial amount O(n4) [9] of feasible cliquesK ′, and all of them can be easily obtained from the initial K arisen with thenecessary condition. Line 3 represents this important step. If Algorithm 1returns a (2, 1) probe partition at Line 6, then such partition is formed by abipartite graph (Line 5) and a probe clique S ∪ K ′ (Line 4); or at Line 11,such partition is formed by the bipartite graph G[P ] (Line 1) and the probeclique N . If Algorithm 1 returns NO, then at Line 13 the necessary conditionfailed, or at Line 9 the bipartite subgraph could not be obtained.

Algorithm 1 (2, 1) Partitioned Probe Recognition Algorithm

1: if G[P ] admits a (2, 1) partition (I1, I2, K) then2: if K �= ∅ then3: for every clique K ′ of G[P ] where G[P ] \K ′ is bipartite do4: S ← {v ∈ N |uv ∈ E, ∀u ∈ K ′}5: if G \ (S ∪K ′) is bipartite then6: return K = K ′ ∪ S; (I1, I2) = V \ (S ∪K ′)7: end if8: end for9: return NO

10: end if11: return K = N ; (I1, I2) = P12: end if13: return NO

By a similar method, (1, 2) partitioned probe problem is in P. Weobserve that (2, 1) and (1, 2) partitioned probe are not complementaryproblems: all edges that must be added in the (2, 1) probe partition are insideof one clique whereas in (1, 2) version, these edges can be inside of two cliques.

S. Dantas et al. / Electronic Notes in Discrete Mathematics 44 (2013) 39–4542

3 NP-complete problem: (2, 2) partitioned probe

We show a polynomial reduction from an instance I = (U,C) with |U | =n and |C| = m, of the NP-complete problem 3-sat to a special instance(G = (V,E), (N,P )) of (2, 2) partitioned probe problem.

Graph G = (V,E) is made up with three types of subgraphs: 1. the basesubgraph Bs; 2. for each variable ui of U a subgraph Truth Setting Ti of G;and 3. for each clause cj = (r, s, t) of C a subgraph Satisfaction Testing Sj.

The Bs subgraph satisfies V (Bs) = A ∪ B ∪ C ∪ D, such that A ={a1, a2, a3, a4, a5, a6}, B = {b1, b2, b3, b4, b5, b6}, C = {c1, c2, c3, c4, c5, c6}, andD = {d1, d2, d3, d4, d5, d6}, where A and B are independent sets, and C andD are complete sets of G. Between vertices of A,B,C and D there arethe following set of edges: EAB = {aibj : i, j ∈ {1, 2, 3, . . . , 6}}, EAC ={aicj : i ∈ {1, 2, 3, . . . , 6}, j ∈ {i, ((i + 1) mod 6) + 1, ((i + 2) mod 6) + 1}},EAD = {aidj : i ∈ {1, 2, 3, . . . , 6}, j ∈ {i, ((i + 1) mod 6) + 1, ((i + 2) mod6) + 1}}, EBC = {bicj : i ∈ {1, 2, 3, . . . , 6}, j ∈ {i, ((i+ 1) mod 6) + 1, ((i+ 2)mod 6) + 1}}, EBD = {bidj : i ∈ {1, 2, 3, . . . , 6}, j ∈ {i, ((i + 1) mod6)+ 1, ((i+2) mod 6)+ 1}}, ECD = ∅. Thus, E(Bs) = ∪i �=j;i,j∈{A,B,C,D}E(ij).The Ti subgraph is described by the vertex set V (Ti) = {ui, ui}. There isa set of edges EU = {uiuj, uiuj, uiuj, uiuj : i �= j; i, j ∈ {1, 2, 3, . . . , n}}∪{uiaj, uibj, uicj, uidj, uiaj, uibj, uicj, uidj : i ∈ {1, 2, 3, . . . , n}, j ∈ {1, 2, 3, 4,5, 6}}. The Sj subgraph satisfies V (Sj) = { rj(1), rj(2), rj(3), sj(1), sj(2),sj(3), tj(1), tj(2), tj(3)}. There is a set of edgesE(Sj) = {rj(1)rj(2), rj(2)rj(3),rj(3)sj(1), sj(1)sj(2), sj(2)sj(3), sj(3)tj(1), tj(1)tj(2), tj(2)tj(3), tj(3)rj(1)}⋃ {rj(1)di, sj(1)di, tj(1)di : i ∈ {1, 2, 3, . . . , n}, j ∈ {1, 2, 3, . . . ,m}}.

If cj = (r, s, t), then we add to G the edges Ej = {rj(1)r, rj(1)x, rj(1)x,sj(1)s, sj(1)y, sj(1)y, tj(1)t, tj(1)z, tj(1)z : x, y, z ∈ U , x �= r �= x, y �= s �= y,z �= t �= z} ⋃{dkuj

i (1), dkuij(1) : k ∈ {1, 2, 3, 4, 5, 6}, i ∈ {1, 2, 3, . . . , n}, j ∈

{1, 2, 3, . . . ,m}.Hence, V = V (Bs)∪

⋃ni=1 V (Ti)∪

⋃mj=1 (V (Sj)), and E = E(Bs)∪E(U)∪

⋃ni=1 (E(Sj) ∪ Ej). The partition (P,N) of V satisfies: N = {rj(1), sj(1), tj(1) :

j ∈ {1, 2, 3, . . . ,m}}, and P = V \N .

The key property of the constructed instance to achieve the NP-complete-ness result is: If H = (V,E) is a (2, 2)-probe graph of G with partition (S1, S2,K1, K2), then we can assume that A ⊂ S1, B ⊂ S2, C ⊂ K1, D ⊂ K2; vertexui ∈ C if only if vertex ui ∈ D, associated with a truth assignment.

On the other hand, each Sj is a C9 in G, which forces one of its non probevertices to enter the partition set K2 corresponding to a satisfied clause cj inthe case we have a (2, 2)-probe graph.

S. Dantas et al. / Electronic Notes in Discrete Mathematics 44 (2013) 39–45 43

Concluding full complexity dichotomy

Class Recognition Probe Sandwich

(2, 1) P P NPC

(2, 2) P NPC NPC

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