On the vertex <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi></mml:math>-path cover

Brešar, Boštjan; Jakovac, Marko; Katrenič, Ján; Semanišin, Gabriel; Taranenko, Andrej

doi:10.1016/j.dam.2013.02.024

Cited by 53 publications

(44 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, and so x i is a stem, whose degree is more than 2, a contradiction with Lemma 4. 4. Hence, we infer that also deg(x i ) = 2, and let y i be another neighbor of x i .…”

Section: Uniformly Dissociated Graphs With Girth At Leastmentioning

confidence: 88%

“…, x i,ji are leaves. Hence z i is a stem in G − N [V (M )], which has degree at least 3, a contradiction with Lemma 4 4…”

mentioning

confidence: 81%

“…The dissociation number was introduced by Papadimitriou and Yannakakis [14] in relation with the complexity of the so-called restricted spanning tree problem. Another closely related concept is the k-path vertex cover, which was introduced in [5] and studied in several papers [4,10]; the corresponding invariant, the k-path vertex cover number of an arbitrary graph G, is denoted by ψ k (G). As it turns out, dissociation sets are complements of 3-path vertex covers of G, and so the following relation holds:…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Uniformly dissociated graphs

Brešar

Hartnell²,

Rall

2017

Ars Math. Contemp.

Self Cite

View full text Add to dashboard Cite

A set D of vertices in a graph G is called a dissociation set if every vertex in D has at most one neighbor in D. We call a graph G uniformly dissociated if all maximal dissociation sets are of the same cardinality. Characterizations of uniformly dissociated graphs with small cardinalities of dissociation sets are proven; in particular, the graphs in which all maximal dissociation sets are of cardinality 2 are the complete graphs on at least two vertices from which possibly a matching is removed, while the graphs in which all maximal dissociation sets are of cardinality 3 are the complements of the K 4-free geodetic graphs with diameter 2. A general construction by which any graph can be embedded as an induced subgraph of a uniformly dissociated graph is also presented. In the main result we characterize uniformly dissociated graphs with girth at least 7 to be either isomorphic to C 7 , or obtainable from an arbitrary graph H with girth at least 7 by identifying each vertex of H with a leaf of a copy of P 3 .

show abstract

“…, and so x i is a stem, whose degree is more than 2, a contradiction with Lemma 4. 4. Hence, we infer that also deg(x i ) = 2, and let y i be another neighbor of x i .…”

Section: Uniformly Dissociated Graphs With Girth At Leastmentioning

confidence: 88%

“…, x i,ji are leaves. Hence z i is a stem in G − N [V (M )], which has degree at least 3, a contradiction with Lemma 4 4…”

mentioning

confidence: 81%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Uniformly dissociated graphs

Brešar

Hartnell²,

Rall

2017

Ars Math. Contemp.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The k-path vertex cover problem was heavily studied on several types of graphs products. Already in [4] the Cartesian product of two paths, i.e. grid graphs, was considered.…”

Section: Introductionmentioning

confidence: 99%

The k-path vertex cover: general bounds and chordal graphs

Bujtás¹,

Jakovac²,

Tuza³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

For an integer k ≥ 3, a k-path vertex cover of a graph G = (V, E) is a set T ⊆ V that shares a vertex with every path subgraph of order k in G. The minimum cardinality of a k-path vertex cover is denoted by ψ k (G). We give estimates -mostly upper bounds -on ψ k (G) in terms of various parameters, including vertex degrees and the number of vertices and edges. The problem is also considered on chordal graphs and planar graphs.

show abstract

“…Among the two parameters

ν_{k} (G)

and

τ_{k} (G)

, only the latter seems to have received considerable attention in the literature . Note that a set

X

of vertices of a graph

G

is a

3

‐vertex cover if and only if its complement

V (G) / X

is a so‐called dissociation set , that is, a set of vertices inducing a subgraph of maximum degree at most

1

.…”

Section: Introductionmentioning

confidence: 99%

On the Kőnig‐Egerváry theorem for ‐paths

Bessy

Ochem

Rautenbach

2018

Journal of Graph Theory

View full text Add to dashboard Cite

The famous Kőnig‐Egerváry theorem is equivalent to the statement that the matching number equals the vertex cover number for every induced subgraph of some graph if and only if that graph is bipartite. Inspired by this result, we consider the set G k of all graphs such that, for every induced subgraph, the maximum number of disjoint paths of order k equals the minimum order of a set of vertices intersecting all paths of order k. For k ∈ { 3 , 4 }, we give complete structural descriptions of the graphs in G k. Furthermore, for odd k, we give a complete structural description of the graphs in G k that contain no cycle of order less than k. For these graph classes, our results yield efficient recognition algorithms as well as efficient algorithms that determine maximum sets of disjoint paths of order k and minimum sets of vertices intersecting all paths of order k.

show abstract

On the vertex k-path cover

Cited by 53 publications

References 10 publications

Uniformly dissociated graphs

Uniformly dissociated graphs

The k-path vertex cover: general bounds and chordal graphs

On the Kőnig‐Egerváry theorem for ‐paths

Contact Info

Product

Resources

About