An Estimate of the Tree-Width of a Planar Graph Which Has Not a Given Planar Grid as a Minor.

1998 ◽  
pp. 372-383
Author(s):  
K. Yu. Gorbunov
Keyword(s):  
Planar Graph ◽  
Planar Grid ◽  
Tree Width ◽  
A Minor

scholarly journals Excluding any graph as a minor allows a low tree-width 2-coloring

2004 ◽  
Vol 91 (1) ◽  
pp. 25-41 ◽  
Author(s):  
Matt DeVos ◽  
Guoli Ding ◽  
Bogdan Oporowski ◽  
Daniel P. Sanders ◽  
Bruce Reed ◽  
...  
Keyword(s):  
Tree Width ◽  
A Minor

A SIMPLE HEURISTIC FOR MINIMUM CONNECTED DOMINATING SET IN GRAPHS

2003 ◽  
Vol 14 (02) ◽  
pp. 323-333 ◽  
Author(s):  
PENG-JUN WAN ◽  
KHALED M. ALZOUBI ◽  
OPHIR FRIEDER
Keyword(s):  
Planar Graph ◽  
Complete Graph ◽  
Dominating Set ◽  
Independence Number ◽  
Domination Number ◽  
Connected Dominating Set ◽  
Minimum Connected Dominating Set ◽  
Simple Heuristic ◽  
Connected Domination Number ◽  
A Minor

Let α2(G), γ(G) and γc(G) be the 2-independence number, the domination number, and the connected domination number of a graph G respectively. Then α2(G) ≤ γ (G) ≤ γc(G). In this paper , we present a simple heuristic for Minimum Connected Dominating Set in graphs. When running on a graph G excluding Km (the complete graph of order m) as a minor, the heuristic produces a connected dominating set of cardinality at most 7α2(G) - 4 if m = 3, or at most [Formula: see text] if m ≥ 4. In particular, if running on a planar graph G, the heuristic outputs a connected dominating set of cardinality at most 15α2(G) - 5.

scholarly journals Algebraically Grid-Like Graphs have Large Tree-Width

10.37236/7691 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Daniel Weißauer
Keyword(s):  
Real Number ◽  
Long Cycle ◽  
Large Tree ◽  
Tree Width ◽  
A Minor ◽  
Grid Minor ◽  
Large Grid

By the Grid Minor Theorem of Robertson and Seymour, every graph of sufficiently large tree-width contains a large grid as a minor. Tree-width may therefore be regarded as a measure of 'grid-likeness' of a graph. The grid contains a long cycle on the perimeter, which is the $\mathbb{F}_2$-sum of the rectangles inside. Moreover, the grid distorts the metric of the cycle only by a factor of two. We prove that every graph that resembles the grid in this algebraic sense has large tree-width: Let $k, p$ be integers, $\gamma$ a real number and $G$ a graph. Suppose that $G$ contains a cycle of length at least $2 \gamma p k$ which is the $\mathbb{F}_2$-sum of cycles of length at most $p$ and whose metric is distorted by a factor of at most $\gamma$. Then $G$ has tree-width at least $k$.

scholarly journals Coloring with no $2$-Colored $P_4$'s

10.37236/1779 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Michael O. Albertson ◽  
Glenn G. Chappell ◽  
H. A. Kierstead ◽  
André Kündgen ◽  
Radhika Ramamurthi
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Short Proof ◽  
Cubic Graphs ◽  
Proper Coloring ◽  
Be Star ◽  
Tree Width ◽  
Star Coloring

A proper coloring of the vertices of a graph is called a star coloring if every two color classes induce a star forest. Star colorings are a strengthening of acyclic colorings, i.e., proper colorings in which every two color classes induce a forest. We show that every acyclic $k$-coloring can be refined to a star coloring with at most $(2k^2-k)$ colors. Similarly, we prove that planar graphs have star colorings with at most 20 colors and we exhibit a planar graph which requires 10 colors. We prove several other structural and topological results for star colorings, such as: cubic graphs are $7$-colorable, and planar graphs of girth at least $7$ are $9$-colorable. We provide a short proof of the result of Fertin, Raspaud, and Reed that graphs with tree-width $t$ can be star colored with ${t+2\choose2}$ colors, and we show that this is best possible.

scholarly journals Non-Separating Planar Graphs

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.

scholarly journals A tight Erdős-Pósa function for planar minors

2019 ◽  
Author(s):  
Wouter Cames van Batenburg ◽  
Tony Huyn ◽  
Gwenaël Joret ◽  
Jean-Florent Raymond
Keyword(s):  
Planar Graph ◽  
General Theorem ◽  
Large Body ◽  
Minimum Size ◽  
Property A ◽  
Disjoint Cycles ◽  
The Family ◽  
Odd Cycle ◽  
A Minor ◽  
Odd Cycles

Let F be a family of graphs. Then for every graph G the maximum number of disjoint subgraphs of G, each isomorphic to a member of F, is at most the minimum size of a set of vertices that intersects every subgraph of G isomorphic to a member of F. We say that F packs if equality holds for every graph G. Only very few families pack. As the next best weakening we say that F has the Erdős-Pósa property if there exists a function f such that for every graph G and integer k>0 the graph G has either k disjoint subgraphs each isomorphic to a member of F or a set of at most f(k) vertices that intersects every subgraph of G isomorphic to a member of F. The name is motivated by a classical 1965 result of Erdős and Pósa stating that for every graph G and integer k>0 the graph G has either k disjoint cycles or a set of O(klogk) vertices that intersects every cycle. Thus the family of all cycles has the Erdős-Pósa property with f(k)=O(klogk). In contrast, the family of odd cycles fails to have the Erdős-Pósa property. For every integer ℓ, a sufficiently large Escher Wall has an embedding in the projective plane such that every face is even and every homotopically non-trivial closed curve intersects the graph at least ℓ times. In particular, it contains no set of ℓ vertices such that each odd cycle contains at least one them, yet it has no two disjoint odd cycles. By now there is a large body of literature proving that various families F have the Erdős-Pósa property. A very general theorem of Robertson and Seymour says that for every planar graph H the family F(H) of all graphs with a minor isomorphic to H has the Erdős-Pósa property. (When H is non-planar, F(H) does not have the Erdős-Pósa property.) The present paper proves that for every planar graph H the family F(H) has the Erdős-Pósa property with f(k)=O(klogk), which is asymptotically best possible for every graph H with at least one cycle.

scholarly journals Excluded Forest Minors and the Erdős–Pósa Property

2013 ◽  
Vol 22 (5) ◽  
pp. 700-721 ◽  
Author(s):  
SAMUEL FIORINI ◽  
GWENAËL JORET ◽  
DAVID R. WOOD
Keyword(s):  
Positive Integer ◽  
Planar Graph ◽  
Classical Result ◽  
A Minor ◽  
Vertex Disjoint ◽  
Special Case

A classical result of Robertson and Seymour states that the set of graphs containing a fixed planar graph H as a minor has the so-called Erdős–Pósa property; namely, there exists a function f depending only on H such that, for every graph G and every positive integer k, the graph G has k vertex-disjoint subgraphs each containing H as a minor, or there exists a subset X of vertices of G with |X| ≤ f(k) such that G − X has no H-minor (see Robertson and Seymour, J. Combin. Theory Ser. B41 (1986) 92–114). While the best function f currently known is exponential in k, a O(k log k) bound is known in the special case where H is a forest. This is a consequence of a theorem of Bienstock, Robertson, Seymour and Thomas on the pathwidth of graphs with an excluded forest-minor. In this paper we show that the function f can be taken to be linear when H is a forest. This is best possible in the sense that no linear bound is possible if H has a cycle.

scholarly journals Bounding Tree-Width via Contraction on the Projective Plane and Torus.

10.37236/2534 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Evan Morgan ◽  
Bogdan Oporowski
Keyword(s):  
Projective Plane ◽  
Planar Graph ◽  
Tree Width ◽  
Toroidal Graph ◽  
Edge Partition

If $X$ is a collection of edges in a graph $G$, let $G/X$ denote the contraction of $X$. Following a question of Oxley and a conjecture of Oporowski, we prove that every projective planar graph $G$ admits an edge partition $\{X,Y\}$ such that $G/X$ and $G/Y$ have tree-width at most three. We prove that every toroidal graph $G$ admits an edge partition $\{X,Y\}$ such that $G/X$ and $G/Y$ have tree-width at most three and four, respectively.

scholarly journals Tree-width and large grid minors in planar graphs

2011 ◽  
Vol Vol. 13 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Alexander Grigoriev
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Tree Decomposition ◽  
International Audience ◽  
Tree Width ◽  
Grid Minor ◽  
Large Grid

Graphs and Algorithms International audience We show that for a planar graph with no g-grid minor there exists a tree-decomposition of width at most 5g - 6. The proof is constructive and simple. The underlying algorithm for the tree-decomposition runs in O(n(2) log n) time.

Vertex-Bipartition Method for Colouring Minor-Closed Classes of Graphs

2010 ◽  
Vol 19 (4) ◽  
pp. 579-591 ◽  
Author(s):  
GUOLI DING ◽  
STAN DZIOBIAK
Keyword(s):  
Approximation Algorithms ◽  
Absolute Constant ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Fixed Number ◽  
Time Approximation ◽  
Closed Class ◽  
Tree Width ◽  
A Minor ◽  
Minor Free Graphs

Thomas conjectured that there is an absolute constant c such that for every proper minor-closed class of graphs, there is a polynomial-time algorithm that can colour every G ∈ with at most χ(G) + c colours. We introduce a parameter ρ(), called the degenerate value of , which is defined to be the smallest r such that every G ∈ can be vertex-bipartitioned into a part of bounded tree-width (the bound depending only on ), and a part that is r-degenerate. Although the existence of one global bound for the degenerate values of all proper minor-closed classes would imply Thomas's conjecture, we prove that the values ρ() can be made arbitrarily large. The problem lies in the clique sum operation. As our main result, we show that excluding a planar graph with a fixed number of apex vertices gives rise to a minor-closed class with small degenerate value. As corollaries, we obtain that (i) the degenerate value of every class of graphs of bounded local tree-width is at most 6, and (ii) the degenerate value of the class of Kn-minor-free graphs is at most n + 1. These results give rise to P-time approximation algorithms for colouring any graph in these classes within an error of at most 7 and n + 2 of its chromatic number, respectively.

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