7 research outputs found
Edge-Orders
Canonical orderings and their relatives such as st-numberings have been used
as a key tool in algorithmic graph theory for the last decades. Recently, a
unifying concept behind all these orders has been shown: they can be described
by a graph decomposition into parts that have a prescribed vertex-connectivity.
Despite extensive interest in canonical orderings, no analogue of this
unifying concept is known for edge-connectivity. In this paper, we establish
such a concept named edge-orders and show how to compute (1,1)-edge-orders of
2-edge-connected graphs as well as (2,1)-edge-orders of 3-edge-connected graphs
in linear time, respectively. While the former can be seen as the edge-variants
of st-numberings, the latter are the edge-variants of Mondshein sequences and
non-separating ear decompositions. The methods that we use for obtaining such
edge-orders differ considerably in almost all details from the ones used for
their vertex-counterparts, as different graph-theoretic constructions are used
in the inductive proof and standard reductions from edge- to
vertex-connectivity are bound to fail.
As a first application, we consider the famous Edge-Independent Spanning Tree
Conjecture, which asserts that every k-edge-connected graph contains k rooted
spanning trees that are pairwise edge-independent. We illustrate the impact of
the above edge-orders by deducing algorithms that construct 2- and 3-edge
independent spanning trees of 2- and 3-edge-connected graphs, the latter of
which improves the best known running time from O(n^2) to linear time
Convex Grid Drawings of Plane Graphs with Rectangular Contours
In a convex drawing of a plane graph, all edges are drawn as straight-line segments without any edge-intersection and all facial cycles are drawn as convex polygons. In a convex grid drawing, all vertices are put on grid points. A plane graph G has a convex drawing if and only if G is internally triconnected, and an internally triconnected plane graph G has a convex grid drawing on an n × n grid if G is triconnected or the triconnected component decomposition tree T (G) of G has two or three leaves, where n is the number of vertices in G. In this paper, we show that an internally triconnected plane graph G has a convex grid drawing on a 2n × n 2 grid if T (G) has exactly four leaves. We also present an algorithm to find such a drawing in linear time. Our convex grid drawing has a rectangular contour, while most of the known algorithms produce grid drawings having triangular contours
A plane graph representation of triconnected graphs
AbstractGiven a graph G=(V,E), a set S={s1,s2,…,sk} of k vertices of V, and k natural numbers n1,n2,…,nk such that ∑i=1kni=|V|, the k-partition problem is to find a partition V1,V2,…,Vk of the vertex set V such that |Vi|=ni, si∈Vi, and Vi induces a connected subgraph of G for each i=1,2,…,k. For the tripartition problem on a triconnected graph, a naive algorithm can be designed based on a directional embedding of G in the two-dimensional Euclidean space. However, for graphs of large number of vertices, the implementing of this algorithm requires high precision real arithmetic to distinguish two close vertices in the plane. In this paper, we propose an algorithm for dealing with the tripartition problem by introducing a new data structure called the region graph, which represents a kind of combinatorial embedding of the given graph in the plane. The algorithm constructs a desired tripartition combinatorially in the sense that it does not require any geometrical computation with actual coordinates in the Euclidean space
Approximation and inapproximability results on balanced connected partitions of graphs
Graphs and Algorithm
A Linear-Time Algorithm for Four-Partitioning Four-Connected Planar Graphs
Given a graph G=(V,E), four distinct vertices u_1, u_2, u_3, u_4 \in V and four natural numbers n_1, n_2, n_3, n_4 such that \sum_{i=1}^4 n_i=|V|, we wish to find a partition V_1, V_2, V_3, V_4 of the vertex set V such that u_i \in V_i, |V_i|=n_i and V_i induces a conneceted subgraph of G for each 1 \le i \le 4. In this paper we give a simple linear-time algorithm to find such a partition if G is a 4-connected planar graph and u_1, u_2, u_3, u_4 are located on the same face of a plane embedding of G. Our algorithm is based on a "4-canonical decomposition" of G, wich is generalization of an st-numbering and a "canonical 4-ordering" known in the area of graph drawings
A Linear-Time Algorithm for Four-Partitioning Four-Connected Planar Graphs (Extended Abstract)
Given a graph G = (V; E), k distinct vertices u 1 ; u 2 ; 1 1 1, u k 2 V and k natural numbers n 1 ; n 2 ; 1 1 1 ; n k such that P k i=1 n i = jV j, we wish to find a partition V 1 ; V 2 ; 1 1 1 ; V k of the vertex set V such that u i 2 V i , jV i j = n i , and V i induces a connected subgraph of G for each i, 1 i k. Such a partition is called a k- partition of G. The problem of finding a k-partition of a general graph is NP-hard [DF85], and hence it is very unlikely that there is a polynomial-time algorithm to solve the problem. Although not every graph has a k-partition, Gyori and Lov'asz independently proved that every k-connected graph has a k-partition for any u 1 ; u 2 ; 1 1 1 ; u k and n 1 ; n 2 ; 1 1 1 ; n k [G78, L77]. However, their proofs do not yield any polynomial-time algorithm for actually finding a k- ..