Edge-Packing in Planar Graphs

Maximum G Edge-Packing (EPack _G ) is the problem of finding the maximum number of edge-disjoint isomorphic copies of a fixed guest graph G in a host graph H . This paper investigates the computational complexity of edge-packing for planar guests and planar hosts. Edge-packing is solvable in polynomial time when both G and H are trees. Edge-packing is solvable in linear time when H is outerplanar and G is either a 3-cycle or a k -star (a graph isomorphic to K _1,k ). Edge-packing is NP-complete when H is planar and G is either a cycle or a tree with edges. A strategy for developing polynomial-time approximation algorithms for planar hosts is exemplified by a linear-time approximation algorithm that finds a k -star edge-packing of size at least half the optimal. Received May 1995, and in revised form November 1995, and in final form December 1997.     

Drawing Planar Graphs Symmetrically, II: Biconnected Planar Graphs

Symmetry is one of the most important aesthetic criteria in graph drawing because it reveals the structure in the graph. This paper discusses symmetric drawings of biconnected planar graphs. More specifically, we discuss geometric automorphisms, that is, automorphisms of a graph G that can be represented as symmetries of a drawing of G. Finding geometric automorphisms is the first and most difficult step in constructing symmetric drawings of graphs. The problem of determining whether a given graph has a non-trivial geometric automorphism is NP-complete for general graphs. In this paper we present a linear time algorithm for finding planar geometric automorphisms of biconnected planar graphs. A drawing algorithm is also discussed.     

Drawing Planar Graphs Symmetrically, III: Oneconnected Planar Graphs

Symmetry is one of the most important aesthetic criteria in graph drawing because it reveals structure in the graph. This paper discusses symmetric drawings of oneconnected planar graphs. More specifically, we discuss planar (geometric) automorphisms, that is, automorphisms of a graph G that can be represented as symmetries of a planar drawing of G. Finding planar automorphisms is the first and most difficult step in constructing planar symmetric drawings of graphs. The problem of determining whether a given graph has a nontrivial geometric automorphism is NP-complete for general graphs. The two previous papers in this series have discussed the problem of drawing planar graphs with a maximum number of symmetries, for the restricted cases where the graph is triconnected and biconnected. This paper extends the previous results to cover planar graphs that are oneconnected. We present a linear time algorithm for drawing oneconnected planar graphs with a maximum number of symmetries.     

Fast Minor Testing in Planar Graphs

Minor Containment is a fundamental problem in Algorithmic Graph Theory used as a subroutine in numerous graph algorithms. A model of a graph H in a graph G is a set of disjoint connected subgraphs of G indexed by the vertices of H, such that if {u,v} is an edge of H, then there is an edge of G between components C _u and C _v . A graph H is a minor of G if G contains a model of H as a subgraph. We give an algorithm that, given a planar n-vertex graph G and an h-vertex graph H, either finds in time $\mathcal{O}(2^{\mathcal{O}(h)} \cdot n +n^{2}\cdot\log n)$ a model of H in G, or correctly concludes that G does not contain H as a minor. Our algorithm is the first single-exponential algorithm for this problem and improves all previous minor testing algorithms in planar graphs. Our technique is based on a novel approach called partially embedded dynamic programming.     

Bounded-Degree Independent Sets in Planar Graphs

An independent set of a graph is a set of vertices without edges between them. Every planar graph has an independent set of size at least (1/4)n and there are planar graphs for which any independent set has size at most (1/4)n. In this paper similar bounds are provided for the problem of bounded-degree independent set, i.e., an independent set where additionally all vertices have degree less than a pre-specified bound D. Our upper and lower bounds match (up to a small constant) for D 16.     

Dynamically Switching Vertices in Planar Graphs

We consider graphs whose vertices may be in one of two different states: either on or off . We wish to maintain dynamically such graphs under an intermixed sequence of updates and queries. An update may reverse the status of a vertex, by switching it either on or off , and may insert a new edge or delete an existing edge. A query tests whether any two given vertices are connected in the subgraph induced by the vertices that are on . We give efficient algorithms that maintain information about connectivity on planar graphs in O( log ³ n) amortized time per query, insert, delete, switch-on, and switch-off operation over sequences of at least Ω(n) operations, where n is the number of vertices of the graph. Received September 1997; revised January 1999.     

Disk Embeddings of Planar Graphs

Given a planar graph $G=(V,E)$ and a rooted forest ${\FF}=(V_{\FF}, A_{\FF})$ with leaf set $V$, we wish to decide whether $G$ has a plane embedding $\GG$ satisfying the following condition: There are $|V_{\FF}|-|V|$ pairwise noncrossing Jordan curves in the plane one-to-one corresponding to the nonleaf vertices of ${\FF}$ such that for every nonleaf vertex $f$ of ${\FF}$, the interior of the curve $\JJ_f$ corresponding to $f$ contains all the leaf descendants of $f$ in ${\FF}$ but contains no other leaves of ${\FF}$. This problem arises from theoretical studies in geographic database systems. It is unknown whether this problem can be solved in polynomial time. This paper presents an almost linear-time algorithm for a nontrivial special case where the set of leaf descendants of each nonleaf vertex $f$ in ${\FF}$ induces a connected subgraph of $G$.     

Optimal Polygonal Representation of Planar Graphs

In this paper, we consider the problem of representing planar graphs by polygons whose sides touch. We show that at least six sides per polygon are necessary by constructing a class of planar graphs that cannot be represented by pentagons. We also show that the lower bound of six sides is matched by an upper bound of six sides with a linear-time algorithm for representing any planar graph by touching hexagons. Moreover, our algorithm produces convex polygons with edges having at most three slopes and with all vertices lying on an O(n)×O(n) grid.     

Faster Algorithms for All-Pairs Small Stretch Distances in Weighted Graphs

Let G=(V,E) be a weighted undirected graph, with non-negative edge weights. We consider the problem of efficiently computing approximate distances between all pairs of vertices in?G. While many efficient algorithms are known for this problem in unweighted graphs, not many results are known for this problem in weighted graphs. Zwick?(J. Assoc. Comput. Mach. 49:289–317, 2002) showed that for any fixed ε>0, stretch 1+ε distances (a path in G between u,v∈V is said to be of stretch t if its length is at most t times the distance between u and v in G) between all pairs of vertices in a weighted directed graph on n vertices can be computed in $\tilde{O}(n^{\omega})$ time, where ω<2.376 is the exponent of matrix multiplication and n is the number of vertices. It is known that finding distances of stretch less than 2 between all pairs of vertices in G is at least as hard as Boolean matrix multiplication of two n×n matrices. Here we show that all pairs stretch 2+ε distances for any fixed ε>0 in G can be computed in expected time O(n ^9/4). This algorithm uses a fast rectangular matrix multiplication subroutine. We also present a combinatorial algorithm (that is, it does not use fast matrix multiplication) with expected running time O(n ^9/4) for computing all-pairs stretch 5/2 distances in?G. This combinatorial algorithm will serve as a key step in our all-pairs stretch 2+ε distances algorithm.     

Fast 3-coloring Triangle-Free Planar Graphs

Although deciding whether the vertices of a planar graph can be colored with three colors is NP-hard, the widely known Grötzsch’s theorem states that every triangle-free planar graph is 3-colorable. We show the first o(n ²) algorithm for 3-coloring vertices of triangle-free planar graphs. The time complexity of the algorithm is $\mathcal{O}(n\log n)Although deciding whether the vertices of a planar graph can be colored with three colors is NP-hard, the widely known Gr?tzsch’s theorem states that every triangle-free planar graph is 3-colorable. We show the first o(n ²) algorithm for 3-coloring vertices of triangle-free planar graphs. The time complexity of the algorithm is O(nlogn)\mathcal{O}(n\log n) .     

Subgraphs of 4-Regular Planar Graphs

We shall present an algorithm for determining whether or not a given planar graph H can ever be a subgraph of a 4-regular planar graph. The algorithm has running time O(|H|^2.5) and can be used to find an explicit 4-regular planar graph G⊃H if such a graph exists. It shall not matter whether we specify that H and G must be simple graphs or allow them to be multigraphs.     

New Linear-Time Algorithms for Edge-Coloring Planar Graphs

We show efficient algorithms for edge-coloring planar graphs. Our main result is a linear-time algorithm for coloring planar graphs with maximum degree Δ with max {Δ,9} colors. Thus the coloring is optimal for graphs with maximum degree Δ≥9. Moreover for Δ=4,5,6 we give linear-time algorithms that use Δ+2 colors. These results improve over the algorithms of Chrobak and Yung (J. Algorithms 10:35–51, 1989) and of Chrobak and Nishizeki (J. Algorithms 11:102–116, 1990) which color planar graphs using max {Δ,19} colors in linear time or using max {Δ,9} colors in time. R. Cole supported in part by NSF grants CCR0105678 and CCF0515127 and IDM0414763. Ł. Kowalik supported in part by KBN grant 4T11C04425. Part of this work was done while Ł. Kowalik was staying at the Max Planck Institute in Saarbruecken, Germany.     

Sparse Covers for Planar Graphs and Graphs that Exclude a Fixed Minor

We consider the construction of sparse covers for planar graphs and other graphs that exclude a fixed minor. We present an algorithm that gives a cover for the γ-neighborhood of each node. For planar graphs, the cover has radius less than 16γ and degree no more than 18. For every n node graph that excludes a minor of a fixed size, we present an algorithm that yields a cover with radius no more than 4γ and degree O(logn). This is a significant improvement over previous results for planar graphs and for graphs excluding a fixed minor; in order to obtain clusters with radius O(γ), it was required to have the degree polynomial in n. Our algorithms are based on a recursive application of a basic routine called shortest-path clustering, which seems to be a novel approach to the construction of sparse covers. Since sparse covers have many applications in distributed computing, including compact routing, distributed directories, synchronizers, and Universal TSP, our improved cover construction results in improved algorithms for all these problems, for the class of graphs that exclude a fixed minor.     

All Pairs Shortest Distances for Graphs with Small Integer Length Edges

There is a way to transform the All Pairs Shortest Distances (APSD) problem where the edge lengths are integers with small (?M) absolute value into a problem with edge lengths in {−1, 0, 1}. This transformation allows us to use the algorithms we developed earlier ([1]) and yields quite efficient algorithms. In this paper we give new improved algorithms for these problems. Forn=|V| the number of vertices,Mthe bound on edge length, andωthe exponent of matrix multiplication, we get the following results: 1. A directed nonnegative APSD(n, M) algorithm which runs inO(T(n, M)) time, where[formula]2. A undirected APSD(n, M) algorithm which runs inO(M^(ω+1)/2n^ωlog(Mn)) time.     

Maximum Flow in Directed Planar Graphs with Vertex Capacities

In this paper we present an O(nlog n) time algorithm for finding a maximum flow in a directed planar graph, where the vertices are subject to capacity constraints, in addition to the arcs. If the source and the sink are on the same face, then our algorithm can be implemented in O(n) time.     

A Characterization of Planar Graphs by Pseudo-Line Arrangements

Let ?? be an arrangement of pseudo-lines, i.e., a collection of unbounded x -monotone curves in which each curve crosses each of the others exactly once. A pseudo-line graph (??, E) is a graph for which the vertices are the pseudo-lines of ?? and the edges are some unordered pairs of pseudo-lines of ??. A diamond of a pseudo-line graph (??, E) is a pair of edges {p,q} , {p',q'}∈ E , {p,q} ∩ {p',q'}= Ø, such that the crossing point of the pseudo-lines p and q lies vertically between p' and q' and the crossing point of p' and q' lies vertically between p and q . We show that a graph is planar if and only if it is isomorphic to a diamond-free pseudo-line graph. An immediate consequence of this theorem is that the O(k1/3n) upper bound on the k -level complexity of an arrangement of straight lines, which was very recently discovered by Dey, holds for an arrangement of pseudo-lines as well.     

An Algorithm for Straight-Line Drawing of Planar Graphs

We present a new algorithm for drawing planar graphs on the plane. It can be viewed as a generalization of the algorithm of Chrobak and Payne, which, in turn, is based on an algorithm by de Fraysseix, Pach, and Pollack. Our algorithm improves the previous ones in that it does not require a preliminary triangulation step; triangulation proves problematic in drawing graphs ``nicely,' as it has the tendency to ruin the structure of the input graph. The new algorithm retains the positive features of the previous algorithms: it embeds a biconnected graph of n vertices on a grid of size (2n-4) x (n-2) in linear time. We have implemented the algorithm as part of a software system for drawing graphs nicely. Received September 21, 1995; revised March 15, 1996.     

On Triangulating Planar Graphs under the Four-Connectivity Constraint

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. We show that triangulating embedded planar graphs without introducing new separating triangles can be solved in linear time and space. If the initial graph had no separating triangle, the resulting triangulation is 4-connected. If the planar graph is not embedded, then deciding whether there exists an embedding with at most k separating triangles is NP-complete. For biconnected graphs a linear-time approximation which produces an embedding with at most twice the optimal number is presented. With this algorithm we can check in linear time whether a biconnected planar graph can be made 4-connected while maintaining planarity. Several related remarks and results are included. Received August 1, 1995; revised July 8, 1996, and August 23, 1996.