Maximum nullity and zero forcing number on graphs with maximum degree at most three

Alishahi, Meysam; Rezaei-Sani, Elahe; Sharifi, Elahe

doi:10.1016/j.dam.2020.03.027

Cited by 5 publications

(6 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, z r = v 2 be a shortest such path. Let x i be the vertex in N (v i ) \ V (P ), which is not in T , for each i ∈ [2]. If x 2 is not adjacent to a vertex in P , then (a 1 , a 2 , a 3 , v 1 , z 1 , .…”

Section: Cubic Graphs Attaining the N 2 -Boundmentioning

confidence: 99%

“…Let u i ∈ N (v i ) \ {a i } be the neighbor of v i that is not in P . Suppose that u i , i ∈ [2], is not dominated by a vertex from V (P ) \ {v i ). Without loss of generality we may assume that u 2 is not adjacent to any vertex in V (P ) \ {v 2 }.…”

Section: Let U ′ /mentioning

confidence: 99%

“…, z r ) is a Z-sequence, in which each of c, a 1 and v 2 footprints exactly one vertex (notably, v 2 = z r footprints only u 2 ), implying γ Z gr (G) > n 2 , a contradiction. Now, we assume that u i is adjacent to a vertex in V (P ) \ {v i }, for each i ∈ [2]. If u 1 is adjacent to z 1 , then the sequence (a 2 , c, a 1 , v 1 , z 1 ) is a Z-sequence in which each of c, a 1 and z 1 footprints exactly one vertex, which again yields γ Z gr (G) > n 2 .…”

Section: Let U ′ /mentioning

confidence: 99%

“…Several authors considered bounds on the zero forcing number in terms of the order, from which one can directly get dual bounds for the Z-Grundy domination number by using (2). Amos et al [3] proved the general upper bound, Z(G) ≤ (∆−2)n+2 ∆−1 , which holds for all connected graphs G with maximum degree ∆ and order n. Gentner et al [19] characterized the extremal graphs attaining the bound.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Grundy domination and zero forcing in Kneser graphs

2019

View full text Add to dashboard Cite

Given a finite graph G, the maximum length of a sequence (v1,. .. , v k) of vertices in G such that each vi dominates a vertex that is not dominated by any vertex in {v1,. .. , vi−1} is called the Grundy domination number, γgr(G), of G. A small modification of the definition yields the Z-Grundy domination number, which is the dual invariant of the well-known zero forcing number. In this paper, we prove that γgr(G) ≥ n+⌈ k 2 ⌉−2 k−1 holds for every connected k-regular graph of order n different from K k+1 and 2C4. The bound in the case k = 3 reduces to γgr(G) ≥ n 2 , and we characterize the connected cubic graphs with γgr(G) = n 2. If G is different from K4 and K3,3, then n 2 is also an upper bound for the zero forcing number of a connected cubic graph, and we characterize the connected cubic graphs attaining this bound.

show abstract

Section: Cubic Graphs Attaining the N 2 -Boundmentioning

confidence: 99%

Section: Let U ′ /mentioning

confidence: 99%

Section: Let U ′ /mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Grundy domination and zero forcing in Kneser graphs

2019

View full text Add to dashboard Cite

show abstract

“…Zero forcing in graphs was first introduced and studied in an AIM Special Work Group [3] in 2008, and has subsequently been extensively studied in the literature; MathSciNet lists over 140 papers to date on the topic. For a small sample of recent (2020) papers on zero forcing we refer the reader to [4,12,15,16,20]. The notion of total forcing in graphs was first introduced by Davila [7] in 2015 as a strengthening of zero forcing in graphs, and has been studied, for example, in [8][9][10][11].…”

mentioning

confidence: 99%

Zero and total forcing dense graphs

Davila¹,

Henning²,

Pepper³

2023

Discuss. Math. Graph Theory

View full text Add to dashboard Cite

If S is a set of colored vertices in a simple graph G, then one may allow a colored vertex with exactly one non-colored neighbor to force its noncolored neighbor to become colored. If by iteratively applying this color change rule, all of the vertices in G become colored, then S is a zero forcing set of G. The minimum cardinality of a zero forcing set in G, written Z(G), is the zero forcing number of G. If in addition, S induces a subgraph of G without isolated vertices, then S is a total forcing set of G. The total forcing number of G, written F t (G), is the minimum cardinality of a total forcing set in G. In this paper we introduce, and study, the notion of graphs for which all vertices are contained in some minimum zero forcing set, or some minimum total forcing set; we call such graphs ZF-dense and TF-dense, respectively. A graph is ZTF-dense if it is both ZF-dense and TF-dense. We determine various classes of ZTF-dense graphs, including among others, cycles, complete multipartite graphs of order at least three that are not stars, wheels, n-dimensional hypercubes, and diamond-necklaces. We show that no tree of order at least three is ZTF-dense. We show that if G and H are connected graphs of order at least two that are both ZF-dense, then the join G + H of G and H is ZF-dense.

show abstract

On Extremal Graphs for Zero Forcing Number

Liang

2022

Graphs and Combinatorics

View full text Add to dashboard Cite

Maximum nullity and zero forcing number on graphs with maximum degree at most three

Cited by 5 publications

References 17 publications

Grundy domination and zero forcing in Kneser graphs

Grundy domination and zero forcing in Kneser graphs

Zero and total forcing dense graphs

On Extremal Graphs for Zero Forcing Number

Contact Info

Product

Resources

About