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The well-known notion of domination in a graph abstracts the idea of protecting locations with guards. This paper introduces a new graph invariant, the autonomous domination number, which abstracts the idea of defending a collection of locations with autonomous agents following a simple protocol to coordinate their defense using only local information.
We introduce a new bivariate polynomial ${displaystyle J(G; x,y):=sumlimits_{W in V(G)} x^{|W|}y^{|N(W)|}}$ which contains the standard domination polynomial of the graph $G$ in two different ways. We build methods for efficient calculation of this p
The domination polynomials of binary graph operations, aside from union, join and corona, have not been widely studied. We compute and prove recurrence formulae and properties of the domination polynomials of families of graphs obtained by various pr
In combinatorics, a latin square is a $ntimes n$ matrix filled with n different symbols, each occurring exactly once in each row and exactly once in each column. Associated to each latin square, we can define a simple graph called a latin square grap
The power domination problem seeks to find the placement of the minimum number of sensors needed to monitor an electric power network. We generalize the power domination problem to hypergraphs using the infection rule from Bergen et al: given an init
For a graph $G=(V,E)$, we call a subset $ Ssubseteq V cup E$ a total mixed dominating set of $G$ if each element of $V cup E$ is either adjacent or incident to an element of $S$, and the total mixed domination number $gamma_{tm}(G)$ of $G$ is the min