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We investigate the domination game and the game domination number
in the class of split graphs. We prove that
for any isolate-free
-vertex split graph
, thus strengthening the conjectured
general bound and supporting Rall’s
-conjecture. We also characterise split graphs of even order with
The paper introduces a graph theory variation of the general position problem: given a graph
, determine a largest set
of vertices of
such that no three vertices of
lie on a common geodesic. Such a set is a max-gp-set of
and its size is the gp-number
. Upper bounds on
in terms of different isometric covers are given and used to determine the gp-number of several classes of graphs. Connections between general position sets and packings are investigated and used to give lower bounds on the gp-number. It is also proved that the general position problem is NP-complete.
The strong isometric dimension of a graph G is the least number k such that G isometrically embeds into the strong product of k paths. Using Sperner's theorem, the strong isometric dimension of the Hamming graphs K2 □ Kn is determined.
Sierpiński graphs S (n, κ) generalise the Tower of Hanoi graphs—the graph S (n, 3) is isomorphic to the graph Hn of the Tower of Hanoi with n disks. A 1-perfect code (or an efficient dominating set) in a graph G is a vertex subset of G with the property that the closed neighbourhoods of its elements form a partition of V (G). It is proved that the graphs S (n, κ) possess unique 1-perfect codes, thus extending a previously known result for Hn. An efficient decoding algorithm is also presented. The present approach, in particular the proposed (de)coding, is intrinsically different from the approach to Hn.
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