Planarity Allowing Few Error Vertices in Linear Time

Journal of Combinatorial Theory, Series B, 2001

A graph G is an apex graph if it contains a vertex w such that G − w is a planar graph. It is easy to see that the genus g(G) of the apex graph G is bounded above by τ − 1, where τ is the minimum face cover of the neighbors of w, taken over all planar embeddings of G − w. The main result of this paper is the linear lower bound g(G) ≥ τ /160 (if G − w is 3-connected and τ > 1). It is also proved that the minimum face cover problem is NP-hard for planar triangulations and that the minimum vertex cover is NP-hard for 2-connected cubic planar graphs. Finally, it is shown that computing the genus of apex graphs is NP-hard.

2008 49th Annual IEEE Symposium on Foundations of Computer Science, 2008

For every fixed surface S, orientable or non-orientable, and a given graph G, Mohar (STOC'96 and Siam J. Discrete Math. (1999)) described a linear time algorithm which yields either an embedding of G in S or a minor of G which is not embeddable in S and is minimal with this property. That algorithm, however, needs a lot of lemmas which spanned six additional papers. In this paper, we give a new linear time algorithm for the same problem. The advantages of our algorithm are the following: 1. The proof is considerably simpler: it needs only about 10 pages, and some results (with rather accessible proofs) from graph minors theory, while Mohar's original algorithm and its proof occupy more than 100 pages in total. 2. The hidden constant (depending on the genus g of the surface S) is much smaller. It is single exponential in g, while it is doubly exponential in Mohar's algorithm. As a spinoff of our main result, we give another linear time algorithm, which is of independent interest. This algorithm computes the genus and constructs minimum genus embeddings of graphs of bounded tree-width. This resolves a conjecture by Neil Robertson and solves one of the most annoying long standing open question about complexity of algorithms on graphs of bounded tree-width.

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.

Lecture Notes in Computer Science, 1995

A planarizing set of a graph is a set of edges or vertices whose removal leaves a planar graph. It is shown that, if G is an n-vertex graph of maximum degree d and orientable genus g, then there exists a planarizing set of O( p dgn) edges. This result is tight within a constant factor. Similar results are obtained for planarizing vertex sets and for graphs embedded on nonorientable surfaces. Planarizing edge and vertex sets can be found in O(n + g) time, if an embedding of G on a surface of genus g is given.

Proceedings of the twenty-third annual ACM symposium on Theory of computing - STOC '91, 1991

We give an algorithm for imbedding a graph G of n vertices onto an oriented surface of minimal genus g. If g > 0 then we also construct a forbidden subgraph of G which is homeomorphic to a graph of size exp(O(g)!) which cannot be imbedded on a surface of genus g-1. Our algorithm takes sequential time exp(O(g)!)n O(1) . Since exp(O(g)!) = exp(exp(O(glog(g)))), our algorithm is polynomial time for genus g=O(loglog(n)/logloglog(n)). A simple parallel implementation of our algorithm takes parallel time processors. We give also the smallest known upper bound, namely exp(O(g)!), on the number F(g) of homeomorphic distinct forbidden subgraphs for graph imbeddings onto a surface of genus g.

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.

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].

SIAM Journal on Discrete Mathematics, 1996

This paper develops new techniques for constructing separators for graphs embedded on surfaces of bounded genus. For any arbitrarily small positive " we show that any n-vertex graph G of genus g can be divided in O(n + g) time into components whose sizes do not exceed "n by removing a set C of O( p (g + 1=")n) vertices. Our result improves the best previous ones with respect to the size of C and the time complexity of the algorithm. Moreover, we show that one can cut o from G a piece of no more than (1 ?")n vertices by removing a set of O( p n"(g" + 1) vertices. Both results are optimal up to a constant factor.

Lecture Notes in Computer Science, 2011

The problem Cover(H) asks whether an input graph G covers a fixed graph H (i.e., whether there exists a homomorphism G → H which locally preserves the structure of the graphs). Complexity of this problem has been intensively studied. In this paper, we consider the problem PlanarCover(H) which restricts the input graph G to be planar. PlanarCover(H) is polynomially solvable if Cover(H) belongs to P, and it is even trivially solvable if H has no planar cover. Thus the interesting cases are when H admits a planar cover, but Cover(H) is NP-complete. This also relates the problem to the long-standing Negami Conjecture which aims to describe all graphs having a planar cover. Kratochvíl asked whether there are non-trivial graphs for which Cover(H) is NP-complete but PlanarCover(H) belongs to P. We examine the first nontrivial cases of graphs H for which Cover(H) is NP-complete and which admit a planar cover. We prove NP-completeness of PlanarCover(H) in these cases.

Proceedings of the fourtieth annual ACM symposium on Theory of computing - STOC 08, 2008

For every surface S (orientable or non-orientable), we give a linear time algorithm to test the graph isomorphism of two graphs, one of which admits an embedding of face-width at least 3 into S. This improves a previously known algorithm whose time complexity is n O(g) , where g is the genus of S. This is the first algorithm for which the degree of polynomial in the time complexity does not depend on g. The above result is based on two linear time algorithms, each of which solves a problem that is of independent interest. The first of these problems is the following one. Let S be a fixed surface. Given a graph G and an integer k ≥ 3, we want to find an embedding of G in S of facewidth at least k, or conclude that such an embedding does not exist. It is known that this problem is NP-hard when the surface is not fixed. Moreover, if there is an embedding, the algorithm can give all embeddings of face-width at least k, up to Whitney equivalence. Here, the face-width of an embedded graph G is the minimum number of points of G in which some non-contractible closed curve in the surface intersects the graph. In the proof of the above algorithm, we give a simpler proof and a better bound for the theorem by Mohar and Robertson concerning the number of polyhedral embeddings of 3-connected graphs.