Finding k Partially Disjoint Paths in a Directed Planar Graph

A graph is path-pairable if for any pairing of its vertices there exist edge-disjoint paths joining the vertices in each pair. We investigate the behaviour of the maximum degree in path-pairable planar graphs. We show that any $n$-vertex path-pairable planar graph must contain a vertex of degree linear in $n$. Our result generalizes to graphs embeddable on a surface of finite genus.

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Non-Separating Planar Graphs

The Electronic Journal of Combinatorics ◽

10.37236/8816 ◽

2021 ◽

Vol 28 (1) ◽

Author(s):

Hooman R. Dehkordi ◽

Graham Farr

Keyword(s):

Planar Graph ◽

Planar Graphs ◽

Disjoint Union ◽

Disjoint Paths ◽

Outerplanar Graph ◽

Outerplanar Graphs ◽

Structural Characterisation ◽

A Minor ◽

Vertex Disjoint

A graph $G$ is a non-separating planar graph if there is a drawing $D$ of $G$ on the plane such that (1) no two edges cross each other in $D$ and (2) for any cycle $C$ in $D$, any two vertices not in $C$ are on the same side of $C$ in $D$. Non-separating planar graphs are closed under taking minors and are a subclass of planar graphs and a superclass of outerplanar graphs. In this paper, we show that a graph is a non-separating planar graph if and only if it does not contain $K_1 \cup K_4$ or $K_1 \cup K_{2,3}$ or $K_{1,1,3}$ as a minor. Furthermore, we provide a structural characterisation of this class of graphs. More specifically, we show that any maximal non-separating planar graph is either an outerplanar graph or a wheel or it is a graph obtained from the disjoint union of two triangles by adding three vertex-disjoint paths between the two triangles. Lastly, to demonstrate an application of non-separating planar graphs, we use the characterisation of non-separating planar graphs to prove that there are maximal linkless graphs with $3n-3$ edges. Thus, maximal linkless graphs can have significantly fewer edges than maximum linkless graphs; Sachs exhibited linkless graphs with $n$ vertices and $4n-10$ edges (the maximum possible) in 1983.

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Finding k Disjoint Paths in a Directed Planar Graph

SIAM Journal on Computing ◽

10.1137/s0097539792224061 ◽

1994 ◽

Vol 23 (4) ◽

pp. 780-788 ◽

Cited By ~ 52

Author(s):

Alexander Schrijver

Keyword(s):

Planar Graph ◽

Disjoint Paths

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Improved algorithms for finding length-bounded two vertex-disjoint paths in a planar graph and minmax k vertex-disjoint paths in a directed acyclic graph

Journal of Computer and System Sciences ◽

10.1016/j.jcss.2010.01.013 ◽

2010 ◽

Vol 76 (8) ◽

pp. 697-708 ◽

Cited By ~ 4

Author(s):

Chih-Chiang Yu ◽

Chien-Hsin Lin ◽

Biing-Feng Wang

Keyword(s):

Planar Graph ◽

Directed Acyclic Graph ◽

Disjoint Paths ◽

Acyclic Graph ◽

Vertex Disjoint

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Neighbor Sum Distinguishing Total Choosability of IC-Planar Graphs without Theta Graphs Θ2,1,2

Mathematics ◽

10.3390/math9070708 ◽

2021 ◽

Vol 9 (7) ◽

pp. 708

Author(s):

Donghan Zhang

Keyword(s):

Planar Graph ◽

Planar Graphs ◽

Maximum Degree ◽

Total Coloring ◽

Disjoint Paths ◽

Combinatorial Nullstellensatz ◽

Proper Total Coloring

A theta graph Θ2,1,2 is a graph obtained by joining two vertices by three internally disjoint paths of lengths 2, 1, and 2. A neighbor sum distinguishing (NSD) total coloring ϕ of G is a proper total coloring of G such that ∑z∈EG(u)∪{u}ϕ(z)≠∑z∈EG(v)∪{v}ϕ(z) for each edge uv∈E(G), where EG(u) denotes the set of edges incident with a vertex u. In 2015, Pilśniak and Woźniak introduced this coloring and conjectured that every graph with maximum degree Δ admits an NSD total (Δ+3)-coloring. In this paper, we show that the listing version of this conjecture holds for any IC-planar graph with maximum degree Δ≥9 but without theta graphs Θ2,1,2 by applying the Combinatorial Nullstellensatz, which improves the result of Song et al.

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CONSTRAINED POINT-SET EMBEDDABILITY OF PLANAR GRAPHS

International Journal of Computational Geometry & Applications ◽

10.1142/s021819591000344x ◽

2010 ◽

Vol 20 (05) ◽

pp. 577-600 ◽

Cited By ~ 2

Author(s):

EMILIO DI GIACOMO ◽

WALTER DIDIMO ◽

GIUSEPPE LIOTTA ◽

HENK MEIJER ◽

STEPHEN K. WISMATH

Keyword(s):

Planar Graph ◽

General Position ◽

Planar Graphs ◽

Disjoint Paths ◽

Simple Path ◽

Outerplanar Graph ◽

Straight Line ◽

Point Set ◽

Number Of Bends ◽

Set Of Points

This paper starts the investigation of a constrained version of the point-set embed-dability problem. Let G = (V,E) be a planar graph with n vertices, G′ = (V′,E′) a subgraph of G, and S a set of n distinct points in the plane. We study the problem of computing a point-set embedding of G on S subject to the constraint that G′ is drawn with straight-line edges. Different drawing algorithms are presented that guarantee small curve complexity of the resulting drawing, i.e. a small number of bends per edge. It is proved that: • If G′ is an outerplanar graph and S is any set of points in convex position, a point-set embedding of G on S can be computed such that the edges of E\E′ have at most 4 bends each. • If S is any set of points in general position and G′ is a face of G or if it is a simple path, the curve complexity of the edges of E\E′ is at most 8. • If S is in general position and G′ is a set of k disjoint paths, the curve complexity of the edges of E \ E′ is O(2k).

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Computational Micrograph Registration with Sieve Processes

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100164660 ◽

1996 ◽

Vol 54 ◽

pp. 440-441

Author(s):

P.J. Phillips ◽

J. Huang ◽

S. M. Dunn

Keyword(s):

Planar Graph ◽

Efficient Algorithm ◽

Spatial Organization ◽

Spatial Arrangement ◽

Stereo Image ◽

Image Model ◽

Geometric Invariants ◽

Point Set

In this paper we present an efficient algorithm for automatically finding the correspondence between pairs of stereo micrographs, the key step in forming a stereo image. The computation burden in this problem is solving for the optimal mapping and transformation between the two micrographs. In this paper, we present a sieve algorithm for efficiently estimating the transformation and correspondence.In a sieve algorithm, a sequence of stages gradually reduce the number of transformations and correspondences that need to be examined, i.e., the analogy of sieving through the set of mappings with gradually finer meshes until the answer is found. The set of sieves is derived from an image model, here a planar graph that encodes the spatial organization of the features. In the sieve algorithm, the graph represents the spatial arrangement of objects in the image. The algorithm for finding the correspondence restricts its attention to the graph, with the correspondence being found by a combination of graph matchings, point set matching and geometric invariants.

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