A Linear-Time Algorithm to Find Four Independent Spanning Trees in Four-Connected Planar Graphs

1998 ◽  
pp. 310-323 ◽  
Author(s):  
Kazuyuki Miura ◽  
Daishiro Takahashi ◽  
Shin-ichi Nakano ◽  
Takao Nishizeki
Keyword(s):  
Planar Graphs ◽  
Spanning Trees ◽  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Independent Spanning Trees

A LINEAR-TIME ALGORITHM TO FIND FOUR INDEPENDENT SPANNING TREES IN FOUR CONNECTED PLANAR GRAPHS

1999 ◽  
Vol 10 (02) ◽  
pp. 195-210 ◽  
Author(s):  
KAZUYUKI MIURA ◽  
DAISHIRO TAKAHASHI ◽  
SHIN-ICHI NAKANO ◽  
TAKAO NISHIZEKI
Keyword(s):  
Natural Number ◽  
Planar Graph ◽  
Planar Graphs ◽  
Spanning Trees ◽  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Independent Spanning Trees ◽  
Connected Planar Graph

Given a graph G, a designated vertex r and a natural number k, we wish to find k "independent" spanning trees of G rooted at r, that is, k spanning trees such that the k paths connecting r and any vertex v in the k trees are internally disjoint. In this paper we give a linear-time algorithm to find four independent spanning trees in a 4-connected planar graph.

A randomized linear-time algorithm to find minimum spanning trees

1995 ◽  
Vol 42 (2) ◽  
pp. 321-328 ◽  
Author(s):  
David R. Karger ◽  
Philip N. Klein ◽  
Robert E. Tarjan
Keyword(s):  
Spanning Trees ◽  
Linear Time ◽  
Time Algorithm ◽  
Minimum Spanning Trees ◽  
Linear Time Algorithm

LINEAR-TIME ALGORITHMS FOR DISJOINT TWO-FACE PATHS PROBLEMS IN PLANAR GRAPHS

1996 ◽  
Vol 07 (02) ◽  
pp. 95-110 ◽  
Author(s):  
HEIKE RIPPHAUSEN-LIPA ◽  
DOROTHEA WAGNER ◽  
KARSTEN WEIHE
Keyword(s):  
Planar Graphs ◽  
Linear Time ◽  
Time Algorithm ◽  
Disjoint Paths ◽  
Linear Time Algorithm ◽  
Vertex Disjoint ◽  
Linear Time Algorithms

In this paper we present a linear-time algorithm for the vertex-disjoint Two-Face Paths Problem in planar graphs, i.e., the problem of finding k vertex-disjoint paths between pairs of terminals which lie on two face boundaries. The algorithm is based on the idea of finding rightmost paths with a certain property in planar graphs. Using this method, a linear-time algorithm for finding vertex-disjoint paths of a prescribed homotopy is derived. Moreover, the algorithm is modified to solve the more general linkage problem in linear time, as well.

A linear-time algorithm for four-partitioning four-connected planar graphs

1997 ◽  
Vol 62 (6) ◽  
pp. 315-322 ◽  
Author(s):  
Shin-ichi Nakano ◽  
Md.Saidur Rahman ◽  
Takao Nishizeki
Keyword(s):  
Planar Graphs ◽  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm

scholarly journals Convex Grid Drawings of 3-Connected Planar Graphs

1997 ◽  
Vol 07 (03) ◽  
pp. 211-223 ◽  
Author(s):  
Marek Chrobak ◽  
Goos Kant
Keyword(s):  
Planar Graphs ◽  
Linear Time ◽  
Time Algorithm ◽  
Plane Graph ◽  
Linear Time Algorithm ◽  
Convex Polygons ◽  
Line Segments ◽  
Polynomial Size ◽  
Straight Line

We consider the problem of embedding the vertices of a plane graph into a small (polynomial size) grid in the plane in such a way that the edges are straight, nonintersecting line segments and faces are convex polygons. We present a linear-time algorithm which, given an n-vertex 3-connected plane G (with n ≥ 3), finds such a straight-line convex embedding of G into a (n - 2) × (n - 2) grid.

scholarly journals A Linear-Time Algorithm for 4-Coloring Some Classes of Planar Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Zuosong Liang ◽  
Huandi Wei
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Linear Time ◽  
Time Algorithm ◽  
Vertex Coloring ◽  
Computer Assisted ◽  
Linear Time Algorithm ◽  
Four Color Theorem ◽  
Finite Set ◽  
Proper Vertex

Every graph G = V , E considered in this paper consists of a finite set V of vertices and a finite set E of edges, together with an incidence function that associates each edge e ∈ E of G with an unordered pair of vertices of G which are called the ends of the edge e . A graph is said to be a planar graph if it can be drawn in the plane so that its edges intersect only at their ends. A proper k -vertex-coloring of a graph G = V , E is a mapping c : V ⟶ S ( S is a set of k colors) such that no two adjacent vertices are assigned the same colors. The famous Four Color Theorem states that a planar graph has a proper vertex-coloring with four colors. However, the current known proof for the Four Color Theorem is computer assisted. In addition, the correctness of the proof is still lengthy and complicated. In 2010, a simple O n 2 time algorithm was provided to 4-color a 3-colorable planar graph. In this paper, we give an improved linear-time algorithm to either output a proper 4-coloring of G or conclude that G is not 3-colorable when an arbitrary planar graph G is given. Using this algorithm, we can get the proper 4-colorings of 3-colorable planar graphs, planar graphs with maximum degree at most five, and claw-free planar graphs.

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