Graph Planarity by Replacing Cliques with Paths

2010

A graph is a (planar, Kh)-graph if a collection of disjoint clusters can be identified such that the subgraph induced by each cluster is an h-clique and collapsing all clusters yields a planar graph. Recognizing (planar, Kh)-graphs is a special instance of the more general problem of recognizing (X ,Y)-graphs, where X and Y are two chosen families of graphs. This model, together with the (X ,Y)-graph terminology, was introduced in [3] and generalized in [1, 2] to support hybrid visualization. In particular, if the clusters are requested to be a partition of the set of the vertices of the input graph G, as it is in [3], then G is called a strong (X ,Y)-graph, otherwise G is a weak (X ,Y)-graph or, simply, an (X ,Y)-graph. In this paper we address (planar, Kh)-recognition in the weak model, and show that this problem is NP-complete if h ≥ 5. This result parallels the analogous result for the strong model [10, 9]. We remark that allowing the contraction of any clique of size greater th...

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.

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.

The Electronic Journal of Combinatorics, 2018

A path coloring of a graph $G$ is a vertex coloring of $G$ such that each color class induces a disjoint union of paths. We consider a path-coloring version of list coloring for planar and outerplanar graphs. We show that if each vertex of a planar graph is assigned a list of $3$ colors, then the graph admits a path coloring in which each vertex receives a color from its list. We prove a similar result for outerplanar graphs and lists of size $2$.For outerplanar graphs we prove a multicoloring generalization. We assign each vertex of a graph a list of $q$ colors. We wish to color each vertex with $r$ colors from its list so that, for each color, the set of vertices receiving it induces a disjoint union of paths. We show that we can do this for all outerplanar graphs if and only if $q/r \ge 2$. For planar graphs we conjecture that a similar result holds with $q/r \ge 3$; we present partial results toward this conjecture.

Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, 2011

Given an n-vertex graph G, a drawing of G in the plane is a mapping of its vertices into points of the plane, and its edges into continuous curves, connecting the images of their endpoints. A crossing in such a drawing is a point where two such curves intersect. In the Minimum Crossing Number problem, the goal is to find a drawing of G with minimum number of crossings. The value of the optimal solution, denoted by OPT, is called the graph's crossing number. This is a very basic problem in topological graph theory, that has received a significant amount of attention, but is still poorly understood algorithmically. The best currently known efficient algorithm produces drawings with O(log 2 n)• (n + OPT) crossings on bounded-degree graphs, while only a constant factor hardness of approximation is known. A closely related problem is Minimum Planarization, in which the goal is to remove a minimum-cardinality subset of edges from G, such that the remaining graph is planar. Our main technical result establishes the following connection between the two problems: if we are given a solution of cost k to the Minimum Planarization problem on graph G, then we can efficiently find a drawing of G with at most poly(d) • k • (k + OPT) crossings, where d is the maximum degree in G. This result implies an O(n • poly(d) • log 3/2 n)approximation for Minimum Crossing Number, as well as improved algorithms for special cases of the problem, such as, for example, k-apex and bounded-genus graphs.

Discrete Mathematics, 1990

We first show that the removal of 4fi vertices from an n-vertex planar graph with non-negative vertex weights summing to no more than 1 is sufficient to cleave or recursively separate it into components of weight no more than a given E, thus improving on the 2fia bound shown in . We then derive worst-case bounds on the number of vertices necessary to separate a planar graph of a given radius into components of weight no more than E.

IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 1989

Abstmct-In this paper we present two O ( n * ) planarization algorithms-PLANARIZE and MAXIMAL-PLANARIZE. These algorithms are based on Lempel, Even, and Cederbaum's planarity testing algorithm [9] and its implementation using PQ-trees [8]. Algorithm PLANARIZE is for the construction of a spanning planar subgraph of an n-vertex nonplanar graph. This algorithm proceeds by embedding one vertex at a time and, at each step, adds the maximum number of edges possible without creating nonplanarity of the resultant graph. Given a biconnected spanning planar subgraph G,, of a nonplanar graph G, algorithm MAXIMAL-PLANARIZE constructs a maximal planar subgraph of G which contains G,,. This latter algorithm can also be used to maximally planarize a biconnected planar graph.

Computational Geometry, 2015

We initiate the study of the following problem: Given a non-planar graph G and a planar subgraph S of G, does there exist a straight-line drawing Γ of G in the plane such that the edges of S are not crossed in Γ by any edge of G? We give positive and negative results for different kinds of connected spanning subgraphs S of G. Moreover, in order to enlarge the subset of instances that admit a solution, we consider the possibility of bending the edges of G not in S; in this setting we discuss different trade-offs between the number of bends and the required drawing area.

2000

A graph is path k-colorable if it has a vertex k-coloring in which the subgraph induced by each color class is a disjoint union of paths. A graph is path k-choosable if, whenever each vertex is assigned a list of k colors, such a coloring exists in which each vertex receives a color from its list.

Discrete Applied Mathematics, 2004

We consider simple, finite and undirected graphs. Given a graph G. I'(G) denotes its vertex set and n = | l'(G). A complete of G is a subset of E(G) inducing a complete subgraph. A clique is a maximal complete. We also use the terms complete and clique to refer to the corresponding subgraphs. A complete C covers the edge uv if the end vertices, u and r, belong to C. A complete edge cover of G is a family of completes covering all its edges. Given .A = a family of nonempty sets, the sets F, are called members of the family. E is pairwise intersecting if the intersection of any two members is not the 1 Researcher partially supported by FOMEC. E-mail addresses: lilianawjmate.unlp.edu.ar (L. Alcon), marisate'mate.unlp.edu.ar (M. Gutierrez).