Reconfiguration Graphs for Dominating Sets

Adaricheva, Kira; Bozeman, Chassidy; Clarke, Nancy E.; Haas, Ruth; Messinger, Margaret-Ellen; Seyffarth, Karen; Smith, Heather

doi:10.1007/978-3-030-77983-2_6

Cited by 5 publications

(4 citation statements)

References 21 publications

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“…Our development uses a different description of the i-sets which gives shorter proofs and allows use to establish results on the Hamiltonicity of the i-graphs of C n . Results on the Hamiltonicity of other domination reconfiguration problems appears in [1].…”

Section: I-graphsmentioning

confidence: 99%

“…In the first, the i-set (1, 2) becomes (1, 3) after the token slide. In the second, the allowed token slides show the i-set (2, 4) is adjacent to (1,4), (3,4), (2,3), and (2, 5) in I (P 10 ).…”

Section: The I-graph Of P Nmentioning

confidence: 99%

“…To summarize: Fix ∈ {2, 5, 8, ..., 3k − 1}.• If k = 2k , then by(1) and Cases 3 and 4, the subgraphs H 2,5 , ..., H 6k −4,6k −1 of I (C 12k +1 ) all have order 24k + 2, andH 2,5 ∼ = • • • ∼ = H 6k −4,6k −1 ∼ = C 24k +2 .• If k = 2k + 1, then by (2) and Cases 3 and 4, the subgraphs H 2,5 , ..., H 6k −4,6k −1 of I (C 12k +7 ) all have order 24k + 14, andH 2,5 ∼ = • • • ∼ = H 6k −4,6k −1 ∼ = C 24k +14 .However, the subgraph H 6k +2,6k +5 of I (C 12k +7 ) has order 12k +7 and H 6k +2,6k +5 ∼ = C 12k +7 . In Figure11, the subgraph H 8,11 of I (C 19 ) is shown with blue (dotted) edges.…”

mentioning

confidence: 99%

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The $i$-Graphs of Paths and Cycles

Brewster¹,

Mynhardt²,

Teshima³

2023

Preprint

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The independent domination number i(G) of a graph G is the minimum cardinality of a maximal independent set of G, also called an i(G)-set. The i-graph of G, denoted I (G), is the graph whose vertices correspond to the i(G)-sets, and where two i(G)sets are adjacent if and only if they differ by two adjacent vertices. Although not all graphs are i-graph realizable, that is, given a target graph H, there does not necessarily exist a source graph G such that H ∼ = I (G), all graphs have i-graphs. We determine the i-graphs of paths and cycles and, in the case of cycles, discuss the Hamiltonicity of these i-graphs.

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Section: I-graphsmentioning

confidence: 99%

Section: The I-graph Of P Nmentioning

confidence: 99%

mentioning

confidence: 99%

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The $i$-Graphs of Paths and Cycles

Brewster¹,

Mynhardt²,

Teshima³

2023

Preprint

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“…The domination TAR reconfiguration graph of a graph G, denoted by D TAR (G), was studied in [1] after D TAR k (G) was introduced in [12]. The next remark is known.…”

Section: Isomorphisms Of the Tar Graphmentioning

confidence: 99%

Isomorphisms and properties of TAR reconfiguration graphs for zero forcing and other $X$-set parameters

Bong¹,

Carlson²,

Curtis³

et al. 2022

Preprint

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An X-TAR (token addition/removal) reconfiguration graph has as its vertices sets that satisfy some property X, with an edge between two sets if one is obtained from the other by adding or removing one element. This paper considers the X-TAR graph for X− sets of vertices of a base graph G where the X-sets of G must satisfy certain conditions. Dominating sets, power dominating sets, zero forcing sets, and positive semidefinite zero forcing sets are all examples of X-sets. For graphs G and G ′ with no isolated vertices, it is shown that G and G ′ have isomorphic X-TAR reconfiguration graphs if and only if there is a relabeling of the vertices of G ′ such that G and G ′ have exactly the same X-sets. The concept of an X-irrelevant vertex is introduced to facilitate analysis of X-TAR graph isomorphisms. Furthermore, results related to the connectedness of the zero forcing TAR graph are given. We present families of graphs that exceed known lower bounds for connectedness parameters.

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