Matching and l-subgraph contractibility to planar graphs

SIAM Journal on Computing, 1988

The pair (G, D) consisting of a planar graph G V, E) with n vertices together with a subset of d special vertices D V is called k-planar if there is an embedding of G in the plane so that at most k faces of G are required to cover all of the vertices in D. Checking 1-planarity can be done in linear-time since it reduces to a problem of checking planarity of a related graph. We present an algorithm which given a graph G and a value k either determines that G is not k-planar or generates an appropriate embedding and associated minimum cover in O(ckn) time, where c is a constant. Hence, the algorithm runs in linear time for any fixed k. The fact that the time required by the algorithm grows exponentially in k is to be expected since we also show that for arbitrary k, the associated decision problem is strongly NP-complete, even when the planar graph has essentially a unique planar embedding, d 0(n), and all facial cycles have bounded length. These results provide a polynomial-time recognition algorithm for special cases of Steiner tree problems in graphs which are solvable in polynomial time. Key words, complexity, planar graphs, Steiner trees AMS(MOS) subject classifications. 05, 68 1. Introduction. Recently, there has been a great deal of interest in solving the Steiner tree problem in graphs. This problem is NP-complete even for planar grid graphs [GJ1]. (See [GJ2] for an excellent introduction to the area of computational complexity.) So recent work has centered on efficiently-solvable special cases and heuristic methods; see [Wi] for a survey of work on this problem. Throughout this paper we deal with undirected graphs of the form G (V, E), where V is a set of n vertices and E is a set of edges connecting pairs of vertices. A graph is called planar if it can be embedded in the plane. A graph G V, E) together with d special vertices D V is called k-planar if there is a 131anar embedding of G so that at most k faces of G are required to cover all of the vertices in D. Clearly, a planar graph is the same as an n-planar graph. The planarity number of G is the minimum k such that G is k-planar. A recent paper by [EMV] presents an algorithm which solves the Steiner problem in an arbitrary graph; their algorithm runs in polynomial time for k-planar graphs, for any fixed k, with D being the vertices required to be in the Steiner tree. It is easy to see that checking 1-planarity of G V, E) with special vertices D V is equivalent to testing the planarity of the associated graph G*= (V*, E*), where V*= Vt.J {r} and E* E [_J {(r, v)" v D}, and so can be done in linear time [HT2]. They leave as an open question the complexity of testing k-planarity for fixed k->-2. In 2, we present an algorithm which checks to see if a given (G, D) pair is k-planar given a fixed embedding of G and if so, determines the planarity number of G in O(ckn) time, when c is a constant. This is used in 3 to generate an appropriate embedding of G and a cover of D by k or fewer faces, if possible, in O(ckn) time. Hence, the algorithm runs in linear time for any fixed k. The fact that the time required grows exponentially in k is to be expected as we show in 4 that for arbitrary k, the associated decision problem is strongly NP-complete, even when the planar graph has essentially a unique planar embedding, d O(n), and all facial cycles have bounded length.

Discrete Applied Mathematics, 2006

The nonplanar vertex deletion or vertex deletion vd(G) of a graph G is the smallest nonnegative integer k, such that the removal of k vertices from G produces a planar graph G . In this case G is said to be a maximum planar induced subgraph of G. We solve a problem proposed by Yannakakis: find the threshold for the maximum degree of a graph G such that, given a graph G and a nonnegative integer k, to decide whether vd(G) k is NP-complete. We prove that it is NP-complete to decide whether a maximum degree 3 graph G and a nonnegative integer k satisfy vd(G) k. We prove that unless P = NP there is no polynomial-time approximation algorithm with fixed ratio to compute the size of a maximum planar induced subgraph for graphs in general. We prove that it is Max SNP-hard to compute vd(G) when restricted to a cubic input G. Finally, we exhibit a polynomial-time 3 4 -approximation algorithm for finding a maximum planar induced subgraph of a maximum degree 3 graph.

Algorithms

This paper introduces and studies the following beyond-planarity problem, which we call h-Clique2Path Planarity. Let G be a simple topological graph whose vertices are partitioned into subsets of size at most h, each inducing a clique. h-Clique2Path Planarity asks whether it is possible to obtain a planar subgraph of G by removing edges from each clique so that the subgraph induced by each subset is a path. We investigate the complexity of this problem in relation to k-planarity. In particular, we prove that h-Clique2Path Planarity is NP-complete even when h=4 and G is a simple 3-plane graph, while it can be solved in linear time when G is a simple 1-plane graph, for any value of h. Our results contribute to the growing fields of hybrid planarity and of graph drawing beyond planarity.

Journal of Algorithms, 2004

We discuss general techniques, centered around the "Layerwise Separation Property" (LSP) of a planar graph problem, that allow to develop algorithms with running time c √ k |G|, given an instance G of a problem on planar graphs with parameter k. Problems having LSP include planar vertex cover, planar independent set, and planar dominating set. Extensions of our speed-up technique to basically all fixed-parameter tractable planar graph problems are also exhibited. Moreover, we relate, e.g., the domination number or the vertex cover number, with the treewidth of a plane graph. √ k n O(1) for constant c. Moreover, we discuss an extension of our technique which applies to basically all fixed-parameter tractable graph problems.

Discrete Applied Mathematics, 2010

A mapping from the vertex set of a graph G = (V , E) into an interval of integers {0, . . . , k} is an L(2, 1)-labelling of G of span k if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices at distance 2 are mapped onto distinct integers. It is known that, for any fixed k ≥ 4, deciding the existence of such a labelling is an NP-complete problem while it is polynomial for k ≤ 3. For even k ≥ 8, it remains NP-complete when restricted to planar graphs. In this paper, we show that it remains NP-complete for any k ≥ 4 by reduction from Planar Cubic Two-Colourable Perfect

Journal of Combinatorial Theory, Series B, 1981

The object of this paper is to show that every maximal planar graph is recognizable from its family of vertex-deleted subgraphs.

Discrete Mathematics, 2001

Let G be a connected graph with at least 2(m + n + 1) vertices. Then G is E(m; n) if for each pair of disjoint matchings M; N ⊆ E(G) of size m and n, respectively, there exists a perfect matching F in G such that M ⊆ F and F ∩ N = ∅. In the present paper, we wish to study the property E(m; n) for the various values of integers m and n when the graphs in question are restricted to be planar. It is known (Plummer, Annals of Discrete Mathematics 41 (1989) 347-354) that no planar graph is E(3; 0). This result is improved in the present paper by showing that no planar graph is E(2; 1). This severely limits the values of m and n for which a planar graph can have property E(m; n) and leads us to consider the properties E(1; n) and E(0; n) for certain classes of planar graphs. Sharpness of the various results is also explored.

Lecture Notes in Computer Science, 2007

We prove that every triconnected planar graph is definable by a first order sentence that uses at most 15 variables and has quantifier depth at most 11 log 2 n + 43. As a consequence, a canonic form of such graphs is computable in AC 1 by the 14-dimensional Weisfeiler-Lehman algorithm. This provides another way to show that the planar graph isomorphism is solvable in AC 1 . * Supported by an Alexander von Humboldt fellowship.

Electronic Notes in Discrete Mathematics, 2007

Every drawing of a non-planar graph G in the plane induces a planarization, i.e., a planar graph obtained by replacing edge crossings with dummy vertices. In this paper, we consider the relationship between the capacity of a minimum st-cut in a graph G and its planarizations. We show that these capacities need not be equal. On the other hand, we prove that every crossing minimal planarization can be efficiently transformed into another crossing minimal planarization that preserves the capacity of a minimum st-cut in G. Furthermore, we extend the result to general (reasonable) planarizations. This property turns out to be a powerful tool for reducing the computational efforts in crossing minimization algorithms. Another application is the correction of a proof given byŠiráň , that shows an additivity property of the crossing number with respect to certain decompositions.

2009 50th Annual IEEE Symposium on Foundations of Computer Science, 2009

We show that for every fixed k, there is a linear time algorithm that decides whether or not a given graph has a vertex set X of order at most k such that G − X is planar (we call this class of graphs k-apex), and if this is the case, computes a drawing of the graph in the plane after deleting at most k vertices. In fact, in this case, we shall determine the minimum value l ≤ k such that after deleting some l vertices, the resulting graph is planar. If this is not the case, then the algorithm gives rise to a minor which is not kapex and is minimal with this property. This answers the question posed by Cabello and Mohar in 2005, and by Kawarabayashi and Reed (STOC'07), respectively. Note that the case k = 0 is the planarity case. Thus our algorithm can be viewed as a generalization of the seminal result by Hopcroft and Tarjan (J. ACM 1974), which determines if a given graph is planar in linear time. Our algorithm can be also compared to the algorithms by Mohar (STOC'96 and Siam J. Discrete Math 2001) for testing the embeddability of an input graph in a fixed surface in linear time, by Kawarabayashi and Mohar (STOC'08) for testing polyhedral embeddability of an input graph in a fixed surface in linear time, and by Kawarabayashi and Reed (STOC'07) for testing the fixed crossing number in linear time. Note that deciding the genus of k-apex graphs is NP-complete, even for k = 1, as shown by Mohar. Thus k-apex graphs are very different from bounded genus graphs in a sense. In addition, for any fixed c, k, we apply our algorithm to obtain a linear time approximation scheme for weighted TSP, and for minimum weighted c-edge-connected submultigraph, respectively, for k-apex graphs. (In this case, an embedding of a k-apex graph is not given in the input). The first result generalizes the recent planar result by Klein (FOCS'05), while the second result generalizes Czumaj et al. (SODA'04). We also extend several optimization results for planar graphs by Baker (J. ACM. 1994) and others to k-apex graphs.

Lecture Notes in Computer Science, 1998

We prove that the problem to get an inclusion minimal elimination ordering can be solved in linear time for planar graphs. The basic structure of the linear time algorithm is as follows. We select a vertex r as maximum and get a rst approximation of a minimal elimination ordering considering a vertex x as smaller than y if x has a larger distance than y from r. Using planarity, one can determine the ll-in edges joining two vertices of the same distance from r almost immediately. The algorithm determines an O(n)-representation of these ll-in edges. To determine the nal ll-in ordering, we use similar techniques as in the general parallel minimal elimination algorithm of 5].

Algorithmica, 1997

Triangulation of planar graphs under constraints is a fundamental problem in the representation of objects. Related keywords are graph augmentation from the field of graph algorithms and mesh generation from the field of computational geometry. We consider the triangulation problem for planar graphs under the constraint to satisfy 4-connectivity. A 4-connected planar graph has no separating triangles, i.e., cycles of length 3 which are not a face.

Lecture Notes in Computer Science, 2014

A graph G covers a graph H if there exists a locally bijective homomorphism from G to H. We deal with regular covers in which this locally bijective homomorphism is prescribed by an action of a subgroup of Aut(G). Regular covers have many applications in constructions and studies of big objects all over mathematics and computer science.

2008

Planar graph canonization is known to be hard for L this directly follows from L-hardness of tree-canonization [Lin92]. We give a log-space algorithm for planar graph canonization. This gives completeness for log-space under AC 0 many-one reductions and improves the previously known upper bound of AC 1 [MR91]. A planar graph can be decomposed into biconnected components. We give a log-space procedure for the decomposition of a biconnected planar graph into a triconnected component tree. The canonization process is based on these decomposition steps. 1

Lecture Notes in Computer Science, 2011

An important step in laying out hierarchical network diagrams is to order the nodes on each level. The usual approach is to minimize the number of edge crossings. This problem is NP-hard even for two layers when the first layer is fixed. Hence, in practice crossing minimization is performed using heuristics. Another suggested approach is to maximize the planar subgraph, i.e. find the least number of edges to delete to make the graph planar. Again this is performed using heuristics since minimal edge deletion for planarity is NP-hard. We show that using modern SAT and MIP solving approaches we can find optimal orderings for minimal crossing or minimal edge deletion for planarization on reasonably sized graphs. These exact approaches provide a benchmark for measuring quality of heuristic crossing minimization and planarization algorithms. Furthermore, we can straightforwardly extend our approach to minimize crossings followed by maximizing planar subgraph or vice versa; these hybrid approaches produce noticeably better layout then either crossing minimization or planarization alone.