5 research outputs found
On the coalition number of trees
Let be a graph with vertex set and of order , and let
and be the minimum and maximum degree of ,
respectively. Two disjoint sets form a coalition in
if none of them is a dominating set of but their union is. A
vertex partition of is a coalition partition of
if every set is either a dominating set of with the
cardinality , or is not a dominating set but for some ,
and form a coalition. The maximum cardinality of a coalition
partition of is the coalition number of . Given a
coalition partition of , a coalition graph
\CG(G, \Psi) is associated on such that there is a one-to-one
correspondence between its vertices and the members of , where two
vertices of \CG(G, \Psi) are adjacent if and only if the corresponding sets
form a coalition in . In this paper, we partially solve one of the open
problems posed in Haynes et al. \cite{coal0} and we solve two open problems
posed by Haynes et al. \cite{coal1}. We characterize all graphs with
and , and we characterize all trees
with . We determine the number of coalition graphs that can
be defined by all coalition partitions of a given path. Furthermore, we show
that there is no universal coalition path, a path whose coalition partitions
defines all possible coalition graphs
Cops and robber on subclasses of -free graphs
The game of cops and robber is a turn based vertex pursuit game played on a
connected graph between a team of cops and a single robber. The cops and the
robber move alternately along the edges of the graph. We say the team of cops
win the game if a cop and the robber are at the same vertex of the graph. The
minimum number of cops required to win in each component of a graph is called
the cop number of the graph. Sivaraman [Discrete Math. 342(2019), pp.
2306-2307] conjectured that for every , the cop number of a connected
-free graph is at most , where denotes a path on ~vertices.
Turcotte [Discrete Math. 345 (2022), pp. 112660] showed that the cop number of
any -free graph is at most , which was earlier conjectured by
Sivaraman and Testa. Note that if a connected graph is -free, then it is
also -free. Liu showed that the cop number of a connected (,
)-free graph is at most , where is a cycle of length at most or
a claw. So the conjecture of Sivaraman is true for (, )-free graphs,
where is a cycle of length at most or a claw. In this paper, we show
that the cop number of a connected ()-free graph is at most , where
, , diamond, paw, , , ,