A bound on the dissociation number

The dissociation number diss ( G ) $\text{diss}(G)$ of a graph G $G$ is the maximum order of a set of vertices of G $G$ inducing a subgraph that is of maximum degree at most 1. Computing the dissociation number of a given graph is algorithmically hard even when restricted to subcubic bipartite graph...

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Bibliographic Details
Published inJournal of graph theory Vol. 103; no. 4; pp. 661 - 673
Main Authors Bock, Felix, Pardey, Johannes, Penso, Lucia D., Rautenbach, Dieter
Format Journal Article
LanguageEnglish
Published Hoboken Wiley Subscription Services, Inc 01.08.2023
Subjects
Apexes
cactus
dissociation set
Graph theory
Graphs
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ISSN0364-9024
1097-0118
DOI10.1002/jgt.22940

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Summary:The dissociation number diss ( G ) $\text{diss}(G)$ of a graph G $G$ is the maximum order of a set of vertices of G $G$ inducing a subgraph that is of maximum degree at most 1. Computing the dissociation number of a given graph is algorithmically hard even when restricted to subcubic bipartite graphs. For a graph G $G$ with n $n$ vertices, m $m$ edges, k $k$ components, and c 1 ${c}_{1}$ induced cycles of length 1 modulo 3, we show diss ( G ) ≥ n − 1 3 ( m + k + c 1 ) $\text{diss}(G)\ge n-\frac{1}{3}(m+k+{c}_{1})$. Furthermore, we characterize the extremal graphs in which every two cycles are vertex‐disjoint.
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ISSN:0364-9024
1097-0118
DOI:10.1002/jgt.22940