Zero forcing number of fuzzy graphs with application

We introduce and study forcing number for fuzzy graphs. Also, we compute zero forcing numbers for some classes of graphs and extend this concept to fuzzy graphs. In this regard we obtain upper bounds for zero forcing of some classes of fuzzy graphs. We will proceed to obtain a new algorithm to computing zero forcing set and finding a formula for zero forcing number, and by some examples we illustrate these notions. Finally, we introduce some applications of fuzzy zero forcing in medical treatments.

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The Bipartite Zero Forcing Set for a Full Sign Pattern Matrix

Mathematics ◽

10.3390/math8030354 ◽

2020 ◽

Vol 8 (3) ◽

pp. 354

Author(s):

Gu-Fang Mou ◽

Tian-Fei Wang ◽

Zhong-Shan Li

Keyword(s):

Sign Pattern ◽

Zero Forcing ◽

Minimum Rank ◽

Zero Forcing Number ◽

Maximum Nullity ◽

New Variant ◽

Forcing Number ◽

Pattern Matrix ◽

Forcing Set ◽

Zero Forcing Set

For an m × n sign pattern P, we define a signed bipartite graph B ( U , V ) with one set of vertices U = { 1 , 2 , … , m } based on rows of P and the other set of vertices V = { 1 ′ , 2 ′ , … , n ′ } based on columns of P. The zero forcing number is an important graph parameter that has been used to study the minimum rank problem of a matrix. In this paper, we introduce a new variant of zero forcing set−bipartite zero forcing set and provide an algorithm for computing the bipartite zero forcing number. The bipartite zero forcing number provides an upper bound for the maximum nullity of a square full sign pattern P. One advantage of the bipartite zero forcing is that it can be applied to study the minimum rank problem for a non-square full sign pattern.

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Zero forcing number of degree splitting graphs and complete degree splitting graphs

Acta Universitatis Sapientiae Mathematica ◽

10.2478/ausm-2019-0004 ◽

2019 ◽

Vol 11 (1) ◽

pp. 40-53

Author(s):

Charles Dominic

Keyword(s):

Color Change ◽

Simple Graph ◽

White Vertex ◽

Zero Forcing ◽

Zero Forcing Number ◽

Forcing Number ◽

Minimum Number ◽

Splitting Graph ◽

Forcing Set ◽

Zero Forcing Set

Abstract A subset ℤ ⊆ V(G) of initially colored black vertices of a graph G is known as a zero forcing set if we can alter the color of all vertices in G as black by iteratively applying the subsequent color change condition. At each step, any black colored vertex has exactly one white neighbor, then change the color of this white vertex as black. The zero forcing number ℤ (G), is the minimum number of vertices in a zero forcing set ℤ of G (see [11]). In this paper, we compute the zero forcing number of the degree splitting graph (𝒟𝒮-Graph) and the complete degree splitting graph (𝒞𝒟𝒮-Graph) of a graph. We prove that for any simple graph, ℤ [𝒟𝒮(G)] k + t, where ℤ (G) = k and t is the number of newly introduced vertices in 𝒟𝒮(G) to construct it.

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Edge Forcing in Butterfly Networks

Fundamenta Informaticae ◽

10.3233/fi-2021-2074 ◽

2021 ◽

Vol 182 (3) ◽

pp. 285-299

Author(s):

G. Jessy Sujana ◽

T.M. Rajalaxmi ◽

Indra Rajasingh ◽

R. Sundara Rajan

Keyword(s):

Lower Bound ◽

Upper Bound ◽

Higher Dimensions ◽

Zero Forcing ◽

Minimum Cardinality ◽

Forcing Number ◽

Np Complete ◽

Forcing Set ◽

Zero Forcing Set

A zero forcing set is a set S of vertices of a graph G, called forced vertices of G, which are able to force the entire graph by applying the following process iteratively: At any particular instance of time, if any forced vertex has a unique unforced neighbor, it forces that neighbor. In this paper, we introduce a variant of zero forcing set that induces independent edges and name it as edge-forcing set. The minimum cardinality of an edge-forcing set is called the edge-forcing number. We prove that the edge-forcing problem of determining the edge-forcing number is NP-complete. Further, we study the edge-forcing number of butterfly networks. We obtain a lower bound on the edge-forcing number of butterfly networks and prove that this bound is tight for butterfly networks of dimensions 2, 3, 4 and 5 and obtain an upper bound for the higher dimensions.

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All Graphs with a Failed Zero Forcing Number of Two

Symmetry ◽

10.3390/sym13112221 ◽

2021 ◽

Vol 13 (11) ◽

pp. 2221

Author(s):

Luis Gomez ◽

Karla Rubi ◽

Jorden Terrazas ◽

Darren A. Narayan

Keyword(s):

Zero Forcing ◽

The Past ◽

Zero Forcing Number ◽

Forcing Number ◽

Repeated Applications ◽

Zero Forcing Numbers

Given a graph G, the zero forcing number of G, Z(G), is the smallest cardinality of any set S of vertices on which repeated applications of the forcing rule results in all vertices being in S. The forcing rule is: if a vertex v is in S, and exactly one neighbor u of v is not in S, then u is added to S in the next iteration. Zero forcing numbers have attracted great interest over the past 15 years and have been well studied. In this paper, we investigate the largest size of a set S that does not force all of the vertices in a graph to be in S. This quantity is known as the failed zero forcing number of a graph and will be denoted by F(G). We present new results involving this parameter. In particular, we completely characterize all graphs G where F(G)=2, solving a problem posed in 2015 by Fetcie, Jacob, and Saavedra.

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