Graph Sandwich Problems
ROn Shamir1995, Journal of Algorithms
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
The graph sandwich problem for property is de ned as follows: Given two graphs G
Related papers
The graph sandwich problem for 1-join composition is NP-complete
Discrete Applied Mathematics, 2002
A graph is a 1-join composition if its vertex set can be partitioned into four nonempty sets AL; AR; SL and SR such that: every vertex of AL is adjacent to every vertex of AR; no vertex of SL is adjacent to vertex of AR ∪ SR; no vertex of SR is adjacent to a vertex of AL ∪ SL. The graph sandwich problem for 1-join composition is deÿned as follows: Given a vertex set V , a forced edge set E 1 , and a forbidden edge set E 3 , is there a graph G = (V; E) such that E 1 ⊆ E and E ∩ E 3 = ∅, which is a 1-join composition graph? We prove that the graph sandwich problem for 1-join composition is NP-complete. This result stands in contrast to the case where SL = ∅ (SR = ∅), namely, the graph sandwich problem for homogeneous set, which has a polynomial-time solution. ?
Chordal- (k,ℓ)and strongly chordal- (k,ℓ)graph sandwich problems
Journal of the Brazilian Computer Society, 2014
Background: In this work, we consider the graph sandwich decision problem for property , introduced by Golumbic, Kaplan and Shamir: given two graphs G 1 = (V, E 1 ) and G 2 = (V, E 2 ), the question is to know whether there exists a graph G = (V, E) such that E 1 ⊆ E ⊆ E 2 and G satisfies property . Particurlarly, we are interested in fully classifying the complexity of this problem when we look to the following properties : 'G is a chordal-(k, )-graph' and 'G is a strongly chordal-(k, )-graph', for all k, . Methods: In order to do that, we consider each pair of positive values of k and , exhibiting correspondent polynomial algorithms, or NP-complete reductions. Results: We prove that the STRONGLY CHORDAL-(k, ) GRAPH SANDWICH PROBLEM is NP-complete, for k ≥ 1 and ≥ 1, and that the CHORDAL-(k, ) GRAPH SANDWICH PROBLEM is NP-complete, for positive integers k and such that k + ≥ 3. Moreover, we prove that both problems are in P when k or is zero and k + ≤ 2. Conclusions: To complete the complexity dichotomy concerning these problems for all nonnegative values of k and , there still remains the open question of settling the complexity for the case k + ≥ 3 and one of them is equal to zero.
Cograph-( k , ` ) Graph Sandwich Problem
2015
A cograph is a graph without induced P4 (where P4 denotes an induced path with 4 vertices). A graph G is (k, `) if its vertex set can be partitioned into at most k independent sets and ` cliques. Threshold graphs are cographs-(1, 1). Cographs-(2, 1) are a generalization of threshold graphs and, as threshold graphs, they can be recognized in linear time. GRAPH SANDWICH PROBLEMS FOR PROPERTY Π (Π-SP) were defined by Golumbic et al. as a natural generalization of RECOGNITION PROBLEMS. In this paper we show that, although COGRAPH-SP and THRESHOLDSP are polynomially solvable problems, COGRAPH-(2, 1)-SP and JOIN OF TWO THRESHOLDS-SP are NP-complete problems. As a corollary, we have that COGRAPH-(1, 2)-SP is NP-complete as well. Using these results, we prove that COGRAPH-(k, `) is NP-complete for k, ` positive integers such that k + ` ≥ 3.
A General Method for Forbidden Induced Subgraph Sandwich Problem NP-completeness
Electronic Notes in Theoretical Computer Science, 2019
We consider the sandwich problem, a generalization of the recognition problem introduced by Golumbic and Shamir (1993), with respect to classes of graphs defined by excluding induced subgraphs. The Π graph sandwich problem asks, for a pair of graphs G 1 = (V, E 1) and G 2 = (V, E 2) with E 1 ⊆ E 2 , whether there exists a graph G = (V, E) with E 1 ⊆ E ⊆ E 2 that satisfies property Π. We consider the property of being H-free, where H is a fixed graph. Using a new variant of the SAT problem, we present a general framework to establish the NP-completeness of the sandwich problem for several H-free graph classes which generalizes the previous strategy for the class of Hereditary clique-Helly graphs. We also provide infinite families of 3-connected special forbidden induced subgraphs for which each forbidden induced subgraph sandwich problem is NP-complete.
On the forbidden induced subgraph sandwich problem
Discrete Applied Mathematics, 2011
We consider the sandwich problem, a generalization of the recognition problem introduced by Golumbic et al. (1995) [15], with respect to classes of graphs defined by excluding induced subgraphs. We prove that the sandwich problem corresponding to excluding a chordless cycle of fixed length k is NP-complete. We prove that the sandwich problem corresponding to excluding K r \ e for fixed r is polynomial. We prove that the sandwich problem corresponding to 3PC (•, •)-free graphs is NP-complete. These complexity results are related to the classification of a long-standing open problem: the sandwich problem corresponding to perfect graphs.
Total exchange algorithms on ‘sandwich graphs’
Computers & Mathematics with Applications, 1991
Complexity issues for the sandwich homogeneous set problem
Discrete Applied Mathematics, 2011
Keywords: Graph sandwich problems Homogeneous sets Complexity of enumeration problems a b s t r a c t Graph sandwich problems were introduced by Golumbic et al. (1994) in [12] for DNA physical mapping problems and can be described as follows. Given a property Π of graphs and two disjoint sets of edges E 1 , E 2 with E 1 ⊆ E 2 on a vertex set V , the problem is to find a graph G on V with edge set E s having property Π and such that E 1 ⊆ E s ⊆ E 2 .
On Some Classes of Problems on Graphs
2019
The relevance to research the complexity of resolving Graph Theory problems is caused by its numerous applications. In the given paper this problem is investigated in terms of space complexity of data structures that represent analyzed graphs, orgraphs, and directed graphs. The following two non-trivial the simplest sets of problems of Graph Theory are investigated in detail. The first set consists of the problems that can be resolved by some algorithm with space complexity linear relative to the size of the data structure that represents the analyzed graphs. The second set consists of the following problems, such that the size of the solution significantly exceeds the size of the input data. To resolve the problem some algorithm that operates on space linear relative to the size of the data structure that represents the analyzed graphs can be applied. Besides, this algorithm uses some memory of the same size for sequential generation, one fragment after another, the solution of the...
An algorithm for the acyclic hypergraph sandwich problem
2000
Given two hypergraphs H and H on a common vertex-set, we write H < H if each edge of H is contained in an edge of H . Given H < H , either find an acyclic hypergraph A between them, H < A < H , or claim that there is no such A. This problem is referred to as the Acyclic Hypergraph Sandwich Problem (AHSP). It generalizes the concept of treewidth as follows. Let H = G be a graph, |V (G)| = n, and let H = n k consist of all subsets of V (G) of cardinality k. Then the AHSP is solvable if and only if the treewidth of G is strictly less than k, that is T W (G) ≤ k − 1. Another important special case of the AHSP is H = H k , that is edges of H are the unions of all subfamilies of k edges of H. In this case the AHSP generalizes the hypertreewidth of H. In this paper we suggest a simple general algorithm for the AHSP. Though exponential in the worst case, it runs well for many practically important tests with H = H k .
Some parameters of graph and its complement
Discrete Mathematics, 1987
In this paper, we have discussed the Nordhaus-Gaddum problems for diameter d, girth g, circumference c and edge covering number ill-We have both got the following results. If both G and G are connected, then 4<~d+a~<p+l, 4~<d.a~<2p-2. If there are cycles in both G and ~;, then 6~<g+~<p+3, 9~<g.~<3p. If there are cycles in both G and G and p >~ 6, then p+2<.c+~<.2p, 3(p-1)<~c.~<.p 2. If both G and G have no isolated vertex, then where p is the vertex number, s = min{a + b [ r(a + 1, b + 1) >p} and r(a + 1, b + 1) means the well-known Ramsey number. The graphs considered here are finite, undirected and simple. Let G be a graph, V and E be vertex set and edge set of G. Throughout this paper, we always denote vertex number of G by p, chromatic number by X, achromatic number by if, edge chromatic number by Xx, edge connectivity by )., connectivity by r, domination number by v, diameter by d, ~ by g, circumference by c, independent number by tr, edge independent number by trx, coveting number by fl, edge covering number by fix, the degree of vertex v by d(v) and the corresponding parameter of complement t~ of G by f. The symbols [a] = max{x Ix integer, x <~a}, {a} = min{x Ix integer, x ~>a} are also used. In [10], the famous Nordhaus-Gaddum theorem states 2vrp ~< Z + ~<p + 1, (P + 1~ 2 P~<X'X~<\ 2 ]" Since then the relations of some parameters between a graph and its complement are continuously discussed, they are called Nordhaus-Gaddum problems.
Related papers
( k,? ) -Sandwich Problems: why not ask for special kinds of bread?
2014
In this work, we consider the Golumbic, Kaplan, and Shamir decision sandwich problem for a property Π: given two graphs G = (V,E) and G = (V,E), the question is: Is there a graph G = (V,E) such that E ⊆ E ⊆ E and G satisfies Π? The graph G is called sandwich graph. Note that what matters here is just the “filling” of the sandwich. Our proposal is to try different kinds of “bread” for each chosen special sandwich filling. In other words, we focus on the complexity of sandwich problems when, beforehand, it is known that G satisfies a property Π, i = 1, 2. Let (Π,Π,Π)-sp denote the sandwich problem for property Π when G satisfies Π, called sandwich problem with boundary conditions. When G is not required to satisfy any special property, Π is denoted by ∗. A graph G is (k, ?) if there is a partition of V (G) into k independent sets and ? cliques. It is known that (∗, (k, ?), ∗)-sp is NP-complete, for all k+ ? greater than 2. In order to motivate this new work proposal, in this paper we ...
Helly Property and Sandwich Graphs
Electronic Notes in Discrete Mathematics, 2005
Skew partition sandwich problem is NP-complete
Electronic Notes in Discrete Mathematics, 2009
Sandwich problems generalize graph recognition problems with respect to a property Π. A recognition problem has a graph as input, whereas a sandwich problem has two graphs as input. In a sandwich problem, we look for a third graph, required to satisfy a property Π, whose edge set lies between the edge sets of two given graphs. A skew partition of a graph G = (V, E) is a partition of its vertex set V into four nonempty parts A, B, C, D such that each vertex of part A is adjacent to each vertex of part B, and each vertex of part C is nonadjacent to each vertex of part D. Skew cutset generalizes star cutset which in turn generalizes both homogeneous set and clique cutset. Homogeneous set, clique cutset, star cutset, and skew cutset are decompositions arising in perfect graph theory and the recognition of each decomposition is known to be polynomial. Regarding sandwich problems, it is known that homogeneous set sandwich problem is polynomial, clique cutset sandwich problem is NP-complete, and star cutset sandwich problem is polynomial. We prove that skew partition sandwich problem is NP-complete, establishing an interesting computational complexity non-monotonicity.
Helly property, clique graphs, complementary graph classes, and sandwich problems
Journal of the Brazilian Computer Society, 2008
A sandwich problem for property Π asks whether there exists a sandwich graph of a given pair of graphs which has the desired property Π. Graph sandwich problems were first defined in the context of Computational Biology as natural generalizations of recognition problems. We contribute to the study of the complexity of graph sandwich problems by considering the Helly property and complementary graph classes. We obtain a graph class defined by a finite family of minimal forbidden subgraphs for which the sandwich problem is N P-complete. A graph is clique-Helly when its family of cliques satisfies the Helly property. A graph is hereditary clique-Helly when all of its induced subgraphs are clique-Helly. The clique graph of a graph is the intersection graph of the family of its cliques. The recognition problem for the class of clique graphs was a long-standing open problem that was recently solved. We show that the sandwich problems for the graph classes: clique, clique-Helly, hereditary clique-Helly, and clique-Helly nonhereditary are all N P-complete. We propose the study of the complexity of sandwich problems for complementary graph classes as a mean to further understand the sandwich problem as a generalization of the recognition problem.
On the complexity of the sandwich problems for strongly chordal graphs and chordal bipartite graphs
Theoretical Computer Science, 2007
Golumbic, Kaplan, and Shamir, in their paper [M.C. Golumbic, H. Kaplan, R. Shamir, Graph sandwich problems, J. Algorithms 19 (1995) 449-473] on graph sandwich problems published in 1995, left the status of sandwich problems for strongly chordal graphs and chordal bipartite graphs open. We prove that the sandwich problem for strongly chordal graphs is NP-complete. We also give some comments on the computational complexity of the sandwich problem for chordal bipartite graphs.
Partitions and well-coveredness: The graph sandwich problem
Discrete Mathematics
A graph is well-covered if every maximal independent set is also maximum. A (k, ℓ)-partition of a graph G is a partition of its vertex set into k independent sets and ℓ cliques. A graph is (k, ℓ)-wellcovered if it is well-covered and admits a (k, ℓ)-partition. The recognition of (k, ℓ)-well-covered graphs is polynomial-time solvable for the cases (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), and (2, 0), and hard, otherwise. In the Graph Sandwich problem for property Π, we are given a
The polynomial dichotomy for three nonempty part sandwich problems
Electronic Notes in Discrete Mathematics, 2008
We classify into polynomial time or NP -complete all three nonempty part sandwich problems. This solves the polynomial dichotomy for this class of problems.
Bounded Degree Interval Sandwich Problems
1999
The problems of Interval Sandwich (IS) and Intervalizing Colored Graphs (ICG) have received a lot of attention recently, due to their applicability to DNA physical mapping problems with ambiguous data. Most of the results obtained so far on the problems were hardness results. Here we study the problems under assumptions of sparseness, which hold in the biological context. We prove that both problems are polynomial when either (1) the input graph degree and the solution graph clique size are bounded, or (2) the solution graph degree is bounded. In particular, this implies that ICG is polynomial on bounded degree graphs for every xed number of colors, in contrast with the recent result of Bodlaender and de Fluiter. E and F are called the sets of mandatory and forbidden edges, respectively. The graph G 0 , if it exists, is called a sandwich graph for the instance S. Without loss of generality, A Preliminary version of this work was presented at the 4th Israeli Symposium on Theory of Computing and Systems (ISTCS'96) 14]. y AT&T-labs research,
Overview of some solved NP-complete problems in graph theory
In the theory of complexity, NP (nondeterministic polynomial time) is a set of decision problems in polynomial time to be resolved in the nondeterministic Turing machine. Equivalently, it is a set of problems whose solutions can be verified on a deterministic Turing machine in polynomial time. The importance of this class of decision problems is that it contains many interesting problems of search and optimization, where we want to know if there is a solution to the problem. The contents of this paper are now handled NP-complete problems in graph theory.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.