Infection in hypergraphs

Bergen, Ryan; Fallat, Shaun M.; Gorr, Adam; Ihringer, Ferdinand; Meagher, Karen; Purdy, Alison; Yang, Boting; Yu, Guanglong

doi:10.1016/j.dam.2017.11.012

Cited by 6 publications

(45 citation statements)

References 12 publications

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“…. , e ℓ with ℓ ≥ 4 so that e i ∩ e j = ∅ if and only if i − j ≡ ±1 mod ℓ, [3]. In fact, Observation 5.2.4 is true without the restriction on vertex degrees, which follows from the next proposition.…”

Section: Infectious Power Domination Number For Particular Hypergraphsmentioning

confidence: 69%

“…H is a graph (Prop. 1.1 in [3] and page 1097 in [7]). However, Chang and Roussel's definition focuses on allowing one vertex to observe others whereas infectious power domination utilizes the fact that there may be multiple observed vertices in an edge which can be used to observe the edge.…”

Section: Infectious Power Dominationmentioning

confidence: 99%

“…We have the following easy consequence of Observation 3.2. Thus we obtain the several results directly from [3], after some definitions. A hypergraph is an interval hypergraph if there is a linear ordering of the vertices so that each edge consists of consecutive vertices.…”

Section: Infectious Power Domination Number For Particular Hypergraphsmentioning

confidence: 99%

“…First suppose that 1, 3, 5, 7 ∈ V 1 . However, this means that at least two of is a spider cover of T (3) . Thus, sp(T (3) ) = 3.…”

Section: Infectious Power Domination Number For Particular Hypergraphsmentioning

confidence: 99%

“…two graphs) will yield a 4-hypergraph. As the vertex set is the union of the vertex sets of the input hypergraphs, we call this operation a linear sum and use the notation H 1 ⋆ H 2 instead of the term direct product and notation H 1 × H 2 as used in [3]. Theorem 6.3.…”

Section: Linear Sumsmentioning

confidence: 99%

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Infectious power domination of hypergraphs

Bjorkman

2020

Discrete Mathematics

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The power domination problem seeks to find the placement of the minimum number of sensors needed to monitor an electric power network. We generalize the power domination problem to hypergraphs using the infection rule from Bergen et al: given an initial set of observed vertices, S 0 , a set A ⊆ S 0 may infect an edge e if A ⊆ e and for any unobserved vertex v, if A∪ {v} is contained in an edge, then v ∈ e. We combine a domination step with this infection rule to create infectious power domination. We compare this new parameter to the previous generalization by Chang and Roussel. We provide general bounds and determine the impact of some hypergraph operations.

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Section: Infectious Power Domination Number For Particular Hypergraphsmentioning

confidence: 69%

Section: Infectious Power Dominationmentioning

confidence: 99%

Section: Infectious Power Domination Number For Particular Hypergraphsmentioning

confidence: 99%

“…First suppose that 1, 3, 5, 7 ∈ V 1 . However, this means that at least two of is a spider cover of T (3) . Thus, sp(T (3) ) = 3.…”

Section: Infectious Power Domination Number For Particular Hypergraphsmentioning

confidence: 99%

Section: Linear Sumsmentioning

confidence: 99%

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Infectious power domination of hypergraphs

Bjorkman

2020

Discrete Mathematics

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A study of cluster hypergraphs and its properties

Maity¹,

Das²,

Samanta³

et al. 2021

Soc. Netw. Anal. Min.

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In this study, cluster hypergraphs are introduced to generalize the concept of hypergraphs, where cluster nodes are allowed. Few related terms and properties on cluster hypergraphs are discussed. Some operations, including the Cartesian product, union, intersection, etc., are studied. Different types of matrix representations and isomorphism are also proposed on cluster hypergraphs. The notion of an effective degree for nodes is introduced to capture the group/ cluster effects. At last, the area of applications and future directions with conclusions is deployed.

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Zero forcing and maximum nullity for hypergraphs

Hogben

2020

Discrete Applied Mathematics

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The concept of zero forcing is extended from graphs to uniform hypergraphs in analogy with the way zero forcing was defined as an upper bound for the maximum nullity of the family of symmetric matrices whose nonzero pattern of entries is described by a given graph: A family of symmetric hypermatrices is associated with a uniform hypergraph and zeros are forced in a null vector. The value of the hypergraph zero forcing number and maximum nullity are determined for various families of uniform hypergraphs and the effects of several graph operations on the hypergraph zero forcing number are determined. The hypergraph zero forcing number is compared to the infection number of a hypergraph and the iteration process in hypergraph power domination.The edges of a (simple) graph G describe the nonzero off-diagonal pattern of symmetric matrices associated with G. The study of the maximum nullity of these matrices is an active area of research (see [8] and [9] for surveys). The zero forcing number was introduced in [1] as an upper bound for maximum nullity, and independently in control of quantum systems [5] (precise definitions of these and related terms are given in Section 1). As noted in [2], zero forcing is also part of the power domination process on graphs that is used to model the optimal placement of monitoring units in electric networks [11] (see Section 2 for more detail).A natural way to extend the idea of maximum nullity from graphs to hypergraphs is to associate a family of symmetric hypermatrices with a uniform hypergraph. When moving from matrices to hypermatrices there are several possible definitions of rank, nullity, etc. Likewise there are several reasonable choices for how to generalize zero forcing to hypergraphs. In Section 1 we present one standard definition of nullity for hypermatrices and the resulting definition of maximum nullity M 0 (H) of a hypergraph H that generalizes the definition of zero-diagonal maximum nullity M 0 (G) of a graph G introduced in [10]. We

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Infection in hypergraphs

Cited by 6 publications

References 12 publications

Infectious power domination of hypergraphs

Infectious power domination of hypergraphs

A study of cluster hypergraphs and its properties

Zero forcing and maximum nullity for hypergraphs

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