Equitable induced decompositions of twin graphs
Artikel i vetenskaplig tidskrift, 2020

Twins in a graph are vertices that are not adjacent, and which have exactly the
same neighbors. In the twin graph of a graph G, every vertex represents an
equivalence class of twins in G and is weighted by the size of this class.
Motivated by (complex) group testing with two defectives and by fair division
problems, we are interested in decomposing graphs with few twin classes into
k$induced subgraphs with nearly equal edge numbers, where every edge belongs
to exactly one subgraph. Technically we consider the fractional version of the
problem, where the vertices of a weighted twin graph can be split into
arbitrary fractions, and the k induced subgraphs must receive exactly the
same total edge weights. The results then apply to usual graphs, subject to a
small discretization error. We show that such equitable induced decompositions
are indeed possible for various twin graphs, including all bipartite graphs,
cycles Cn and Cn-colorable graphs, and (C3,C5)-free graphs. We also
pay attention to the necessary number of vertices (after the splittings) in the
induced decompositions. Usually this number is bounded by k+O(1), but for
complete bipartite graphs, i.e., when the twin graph is a single edge, roughly
2k^(1/2) vertices suffice, and their exact minimum number is easy to compute
for many k.

induced subgraph decomposition

twins

fractional graph theory

equitable partitioning

complex group testing

Författare

Peter Damaschke

Chalmers, Data- och informationsteknik, Data Science

Australasian Journal of Combinatorics

1034-4942 (ISSN)

Vol. 76 1 24-40

Fundament

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Diskret matematik

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2020-01-10