Relating dissociation, independence, and matchings

Bock, Felix; Pardey, Johannes; Penso, Lúcia Draque; Rautenbach, Dieter

doi:10.1016/j.dam.2022.07.012

Cited by 3 publications

(5 citation statements)

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“…Furthermore, using n p q r s s = + 2 + + + , we obtain s = □ Theorem 7 concerning the extremal graphs for Theorem 1 requires quite some technical preparation, which is why we postpone it to the end of this section. We proceed to the proof of the characterization of the extremal subcubic graphs for (1).…”

Section: Let G Dmentioning

confidence: 99%

“…Section 2. For cubic graphs though, our first main result is the following best possible improvement of (1). Note that the connected cubic graph K 4 satisfies (1) with equality.…”

mentioning

confidence: 91%

“…While ( 1) is trivial, it is NP-hard [1] to recognize the extremal graphs for (1). The dissociation number is algorithmically hard even when restricted, for instance, to subcubic bipartite graphs [3,14,18].…”

mentioning

confidence: 99%

“…The starting point for the present paper were possible improvements of (1). As it turns out, (sub)cubic graphs play a special role in this context, and it is not difficult to construct arbitrarily large connected r-regular extremal graphs for (1) as soon as ≥ r 4, cf.…”

mentioning

confidence: 99%

“…Having considered the influence of degree conditions on (1), we now add further structural conditions. For bipartite graphs, a simple argument yields the following.…”

mentioning

confidence: 99%

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Relating the independence number and the dissociation number

Bock

Pardey

Penso

et al. 2023

Journal of Graph Theory

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The independence number and the dissociation number of a graph are the largest orders of induced subgraphs of of maximum degree at most 0 and at most 1, respectively. We consider possible improvements of the obvious inequality . For connected cubic graphs distinct from , we show , and describe the rich and interesting structure of the extremal graphs in detail. For bipartite graphs, and, more generally, triangle‐free graphs, we also obtain improvements. For subcubic graphs though, the inequality cannot be improved in general, and we characterize all extremal subcubic graphs.

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Section: Let G Dmentioning

confidence: 99%

“…Section 2. For cubic graphs though, our first main result is the following best possible improvement of (1). Note that the connected cubic graph K 4 satisfies (1) with equality.…”

mentioning

confidence: 91%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…Having considered the influence of degree conditions on (1), we now add further structural conditions. For bipartite graphs, a simple argument yields the following.…”

mentioning

confidence: 99%

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