Some Bounds on the Zero Forcing Number of a Graph

Gentner, Michael; Rautenbach, Dieter

doi:10.48550/arxiv.1608.00747

Cited by 6 publications

(14 citation statements)

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“…This minimum degree lower bound has recently been significantly improved when the graph in question has restrictions on its girth and minimum degree. In particular, if G is a graph has minimum degree δ ≥ 2, Genter, Penso, Rautenbach, and Souzab [17] and Gentner and Rautenbach [18] proved F (G) ≥ δ + (δ − 2)(g − 3), whenever g ≤ 6, and also whenever g ≤ 10 by Davila and Henning [12]. Using the techniques presented in [12], Davila, Malinowski, and Stephen [15] recently resolved this inequality in the affirmative for graphs with arbitrary girth.…”

Section: Domination In Graphsmentioning

confidence: 99%

“…In recent years, dynamic colorings of the vertices in a graph has gained much attention. Indeed, forcing sets [1,10,14,17,18,19,25,28], k-forcing sets [2,8], connected forcing sets [6,7,13], and power dominating sets [22,29], have seen a wide verity of application and interesting relationships to other well studied graph properties. These aforementioned sets all share the common property that they may be defined as graph colorings that change during discrete time intervals.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, F (G) and F c (G) have been related to many well studied graph properties such as minimum rank, independence, and domination, see for example [1,2,13]. For more on forcing and connected forcing, we refer the reader to [2,6,7,8,11,12,13,14,17,18].…”

Section: Introductionmentioning

confidence: 99%

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On the total forcing number of a graph

Davila

Henning

2019

Discrete Applied Mathematics

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A total forcing set in a graph G is a forcing set (zero forcing set) in G which induces a subgraph without isolated vertices. Total forcing sets were introduced and first studied by Davila [11]. The total forcing number of G, denoted F t (G) is the minimum cardinality of a total forcing set in G. We study basic properties of F t (G), relate F t (G) to various domination parameters, and establish N P -completeness of the associated decision problem for F t (G). Our main contribution is to prove that if G is a connected graph of order n ≥ 3 with maximum degree ∆, then F t (G) ≤ ( ∆ ∆+1 )n, with equality if and only if G is a complete graph K ∆+1 , or a star K 1,∆ .

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Section: Domination In Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the total forcing number of a graph

Davila

Henning

2019

Discrete Applied Mathematics

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“…A dynamic coloring of the vertices in a graph is a coloring of the vertex set which may change, or propagate, throughout the vertices during discrete time intervals. Of the dynamic colorings, the notion of forcing sets (zero forcing sets), and the associated graph invariant known as the forcing number (zero forcing number ), are arguably the most prominent, see for example [1,10,15,18,19,20,28]. In the study of minimum forcing sets in graphs, it is natural to consider the initial structure of such sets.…”

Section: Introductionmentioning

confidence: 99%

Total forcing sets and zero forcing sets in trees

Davila¹,

Henning²

2020

Discuss. Math. Graph Theory

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A dynamic coloring of the vertices of a graph G starts with an initial subset S of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set S is called a forcing set of G if, by iteratively applying the forcing process, every vertex in G becomes colored. If the initial set S has the added property that it induces a subgraph of G without isolated vertices, then S is called a total forcing set in G. The minimum cardinality of a total forcing set in G is its total forcing number, denoted F t (G). We prove that if T is a tree of order n ≥ 3 with maximum degree ∆, then F t (T ) ≤ 1 ∆ ((∆ − 1)n + 1), and we characterize the infinite family of trees achieving equality in this bound. We also prove that if T is a non-trivial tree with n 1 leaves, then F t (T ) ≥ n 1 , and we characterize the infinite family of trees achieving equality in this bound. As a consequence of this result, the total forcing number of a non-trivial tree is strictly greater than its forcing number. In particular, we prove that if T is a non-trivial tree, then F t (T ) ≥ F (T ) + 1, and we characterize extremal trees achieving this bound.

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“…It is not difficult to show that this bound is attained exactly when G is either K ∆+1 , the complete bipartite graph K ∆,∆ or a cycle. Later, pushing this bound a little further, Gentner and Rautenbach [9] were able to remove the additive constant 2 ∆+1 (for ∆ ≥ 3). Namely, they showed that Z(G) ≤ ∆−2 ∆−1 n holds for every connected graph G of order n and maximum degree ∆ ≥ 3, unless when G is one of five exceptional graphs K ∆+1 , K ∆,∆ , K ∆−1,∆ or two other specific graphs (we do not exhibit them, for full details see [9]).…”

Section: Introductionmentioning

confidence: 99%

On a conjecture of Gentner and Rautenbach

Girão

Mészáros

Smith

2018

Discrete Mathematics

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Gentner and Rautenbach conjectured that the size of a minimum zero forcing set in a connected graph on n vertices with maximum degree 3 is at most 1 3 n + 2. We disprove this conjecture by constructing a collection of connected graphs {Gn} with maximum degree 3 of arbitrarily large order having zero forcing number at least 4 9 |V (Gn)|.time step, the following rule is applied. Namely, if u is a white vertex in G, it will be become black if it is the only white neighbor of some black vertex v, and we say that the vertex v forced u to change color. Once a vertex has been changed to black, it remains black forever. If every vertex of G becomes black in finite time we say that S is a zero forcing set. Throughout this paper, every graph will be finite, simple, and undirected.

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Some Bounds on the Zero Forcing Number of a Graph

Cited by 6 publications

References 0 publications

On the total forcing number of a graph

On the total forcing number of a graph

Total forcing sets and zero forcing sets in trees

On a conjecture of Gentner and Rautenbach

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