Equitable induced decompositions of twin graphs

Journal article, 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.