Volume 57, Number 3, May-June 2023
|Page(s)||1075 - 1086|
|Published online||11 May 2023|
Computing role assignments of Cartesian product of graphs
Instituto de Informática, Universidade Federal de Goiás, Goiás, Brazil
* Corresponding author: email@example.com
Accepted: 3 April 2023
Network science is a growing field of study using Graph Theory as a modeling tool. In social networks, a role assignment is such that individuals play the same role, if they relate in the same way to other individuals playing counterpart roles. In this sense, a role assignment permit to represent the network through a smaller graph modeling its roles. This leads to a problem called r-Role Assignment whose goal is deciding whether it exists such an assignment of r distinct roles. This problem is known to be NP-complete for any fixed r ≥ 2. The Cartesian product of graphs is a well studied graph operation, often used for modeling interconnection networks. Formally, the Cartesian product of G and H is a graph, denoted as G□H, whose vertex set is V(G) × V(H) and two vertices (u, v) and (x, y) are adjacent precisely if u = x and vy ∈ E(H), or ux ∈ E(G) and v = y. In a previous work, we showed that Cartesian product of graphs are always 2-role assignable, however the 3-Role Assignment problem is NP-complete on this class. In this paper, we prove that r-Role Assignment restricted to Cartesian product graphs is still NP-complete, for any fixed r ≥ 4.
Mathematics Subject Classification: 05C78 / 05C76 / 68Q25 / 05C15
Key words: Role assignment / Cartesian product / computational complexity
© The authors. Published by EDP Sciences, ROADEF, SMAI 2023
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