Outerplanar Obstructions for Matroid Pathwidth

Koutsonas, Athanassios; Thilikos, Dimitrios M.; Yamazaki, Koichi

doi:10.1016/j.endm.2011.09.088

Cited by 9 publications

(11 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This is an analogous result to the characterization of acyclic minor obstructions for graphs of path-width at most k, investigated by Takahashi, Ueno, and Kajitani [34], and Ellis, Sudborough, and Turner [19]. As a similar work, Koutsonas, Thilikos, and Yamazaki [26] characterized matroid obstructions for bounded matroid path-width that are cycle matroids of outerplanar graphs.…”

Section: Introductionsupporting

confidence: 65%

Linear rank-width of distance-hereditary graphs II. Vertex-minor obstructions

Kanté

Kwon

2018

European Journal of Combinatorics

View full text Add to dashboard Cite

In the companion paper [Linear rank-width of distance-hereditary graphs I. A polynomialtime algorithm, Algorithmica 78(1):342-377, 2017], we presented a characterization of the linear rank-width of distance-hereditary graphs, from which we derived an algorithm to compute it in polynomial time. In this paper, we investigate structural properties of distance-hereditary graphs based on this characterization.First, we prove that for a fixed tree T , every distance-hereditary graph of sufficiently large linear rank-width contains a vertex-minor isomorphic to T . We extend this property to bigger graph classes, namely, classes of graphs whose prime induced subgraphs have bounded linear rank-width. Here, prime graphs are graphs containing no splits. We conjecture that for every tree T , every graph of sufficiently large linear rank-width contains a vertex-minor isomorphic to T . Our result implies that it is sufficient to prove this conjecture for prime graphs.For a class Φ of graphs closed under taking vertex-minors, a graph G is called a vertex-minor obstruction for Φ if G R Φ but all of its proper vertex-minors are contained in Φ. Secondly, we provide, for each k ě 2, a set of distance-hereditary graphs that contains all distance-hereditary vertex-minor obstructions for graphs of linear rank-width at most k. Also, we give a simpler way to obtain the known vertex-minor obstructions for graphs of linear rank-width at most 1.

show abstract

Section: Introductionsupporting

confidence: 65%

Linear rank-width of distance-hereditary graphs II. Vertex-minor obstructions

Kanté

Kwon

2018

European Journal of Combinatorics

View full text Add to dashboard Cite

show abstract

“…According to [34], if the problem of checking whether p(G) ≤ k is NP-complete, then |O ≤m p,k | is a super-polynomial function of k, unless the polynomial hierarchy collapses to Σ P 3 . Characterizations of p(G) ≤ k (yielding better lower bounds for |O ≤m p,k |) have been provided for several parameters [10,20,40,58,84,98,99,115,117,118]. However, to our knowledge, there is not yet a natural parameter p for which a complete characterization of O ≤m p,k is known.…”

Section: Minor-closed Graph Parametersmentioning

confidence: 99%

Graph Minors and Parameterized Algorithm Design

Thilikos

2012

The Multivariate Algorithmic Revolution and Beyond

Self Cite

View full text Add to dashboard Cite

Abstract. The Graph Minors Theory, developed by Robertson and Seymour, has been one of the most influential mathematical theories in parameterized algorithm design. We present some of the basic algorithmic techniques and methods that emerged from this theory. We discuss its direct meta-algorithmic consequences, we present the algorithmic applications of core theorems such as the grid-exclusion theorem, and we give a brief description of the irrelevant vertex technique.

show abstract

“…In this direction Michael Dinneen proved in [8] that, if all graphs in obs(G) are connected, then |obs(A k (G))| is exponentially big. To show this, Dinneen proved a more general structural theorem claiming that, under the former connectivity assumption, every connected component of a non-connected graph in obs(A k (G)) is a graph in obs(A k (G)), for some k < k. Another way to prove lower bounds to |obs(A k (G))| is to completely characterize, for every k, the set obs(A k (G)) ∩ H, for some graph class H, and then lower bound |obs(A k (G))| by counting (asymptotically or exactly) all the graphs in obs(A k (G)) ∩ H. This last approach has been applied in [28] when G is the class of acyclic graphs and H is the class of outerplanar graphs (see also [13,19]).…”

Section: Introductionmentioning

confidence: 99%

Minor-Obstructions for Apex Sub-unicyclic Graphs

Leivaditis¹,

Singh²,

Stamoulis³

et al. 2019

Preprint

View full text Add to dashboard Cite

A graph is sub-unicyclic if it contains at most one cycle. We also say that a graph G is k-apex sub-unicyclic if it can become sub-unicyclic by removing k of its vertices. We identify 29 graphs that are the minor-obstructions of the class of 1-apex sub-unicyclic graphs, i.e., the set of all minor minimal graphs that do not belong in this class. For bigger values of k, we give an exact structural characterization of all the cactus graphs that are minor-obstructions of k-apex subunicyclic graphs and we enumerate them. This implies that, for every k, the class of k-apex sub-unicyclic graphs has at least 0.34 • k −2.5 (6.278) k minor-obstructions.

show abstract

Outerplanar Obstructions for Matroid Pathwidth

Cited by 9 publications

References 18 publications

Linear rank-width of distance-hereditary graphs II. Vertex-minor obstructions

Linear rank-width of distance-hereditary graphs II. Vertex-minor obstructions

Graph Minors and Parameterized Algorithm Design

Minor-Obstructions for Apex Sub-unicyclic Graphs

Contact Info

Product

Resources

About