Abstract
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. When a smaller graph models the social roles in a network, this gives rise to the decision problem called r-Role Assignment whether it exists such an assignment of r distinct roles to the vertices of the graph. This problem is known to be NP-complete for any fixed r ≥ 2. The Cartesian product of graphs is one of the most studied operation on graphs and has numerous applications in diverse areas, such as Mathematics, Computer Science, Chemistry and Biology. In this paper, we determine the computational complexity of r-Role Assignment restricted to Cartesian product of graphs, for r = 2, 3. In fact, we show that the Cartesian product of graphs is always 2-role assignable, however the problem of 3-Role Assignment is still NP-complete for this class.
Highlights
Graphs have been used for centuries as a modeling tool in which vertices typically represent objects and edges the relationship between them
R-role assignment of a simple graph G is an assignment of r distinct roles to the vertices of G which obeys the rule that two vertices have the same role, if their neighbors have the same set of roles
We have shown that the problem r-Role Assignment restricted to Cartesian product is trivial, with true answer, for r = 2 and NP-complete for r = 3
Summary
Graphs have been used for centuries as a modeling tool in which vertices typically represent objects and edges the relationship between them. We study the problem of r-Role Assignment of Cartesian product in the computational complexity point of view. Whereas role assignment allows to study a network through a smaller graph, Cartesian product is often used for modeling one. Returning to the specific role assignment that correspond to fall coloring, Laskar and Lyle [15] showed that the problem for 3 colors (or 3 roles) is NP-complete even when restricted to the class of bipartite graphs. They construct fall colourable graphs using Cartesian product.
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