5 research outputs found
The Power of Small Coalitions under Two-Tier Majority on Regular Graphs
In this paper, we study the following problem. Consider a setting where a
proposal is offered to the vertices of a given network , and the vertices
must conduct a vote and decide whether to accept the proposal or reject it.
Each vertex has its own valuation of the proposal; we say that is
``happy'' if its valuation is positive (i.e., it expects to gain from adopting
the proposal) and ``sad'' if its valuation is negative. However, vertices do
not base their vote merely on their own valuation. Rather, a vertex is a
\emph{proponent} of the proposal if the majority of its neighbors are happy
with it and an \emph{opponent} in the opposite case. At the end of the vote,
the network collectively accepts the proposal whenever the majority of its
vertices are proponents. We study this problem for regular graphs with loops.
Specifically, we consider the class of -regular graphs
of odd order with all loops and happy vertices. We are interested
in establishing necessary and sufficient conditions for the class
to contain a labeled graph accepting the proposal, as
well as conditions to contain a graph rejecting the proposal. We also discuss
connections to the existing literature, including that on majority domination,
and investigate the properties of the obtained conditions.Comment: 28 pages, 8 figures, accepted for publication in Discrete Applied
Mathematic
Do local majorities force a global majority?
AbstractThe following problem is investigated. Given intrgers r βͺ l βͺ 0 wwith 0, and an integer c for which l β r < c βͺ― l + r, what is the minimum ratio of white balls to black balls over the family of all finite rings of white and black balls that satisfy: (i) the ring has at least one white ball, and (ii) for every white ball, there are at least c more white balls than black balls in the list of the l balls counterclockwise from the white ball conjoined with the list of r balls clockwise from the white ball?Let R(l, r, c) be the minimum ratio and assume that c has the same parity as l + r. For the symmetric cases with l = r = k, it is proved that R(k, k, c) = (2k + c)/(2k β c) when k and 12c are congruent (mod 2), and that R(k, k, c) mightbe slightly larger than (2k + c)/(2k β c) when k and 12c are not congruent. The upper bounds on R given for the latters case are conjectured to be tight.We also conjecture that R(l, r, c) > 1 wheneverc > 0. This is known to be true when l = r. It is shown in general that R(l, r, c) βͺ 1 whenever c > 0 and l β€ 9. Apparently tight upper bounds on R are given for the asymmetric cases with l < r