DOI: 10.1017/s0004972726102159 ISSN: 0004-9727

DOMINATION GAME AND TOTAL DOMINATION GAME PLAYED ON SIERPIŃSKI GRAPHS

TANJA DRAVEC, DANIEL P. JOHNSTON, SANDI KLAVŽAR
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Abstract

The game domination numbers

γ g $\gamma _{\mathrm {g}}$ gamma Subscript normal g
and
γ g ′ $\gamma _{\mathrm {g}}^\prime $ gamma prime Subscript normal g
, and the game total domination numbers
γ tg $\gamma _{\mathrm {{tg}}}$ gamma Subscript tg
and
γ tg ′ $\gamma _{\mathrm {{tg}}}^\prime $ gamma prime Subscript tg
are investigated on Sierpiński graphs
S p n $S_p^n$ upper S Subscript p Superscript n
. For the domination game, the trivial bounds arising from the known domination number and the Grundy domination number of
S p n $S_p^n$ upper S Subscript p Superscript n
are significantly improved by proving that
( 2 p − 3 ) p n − 2 ≤ γ g ( S p n ) , γ g ′ ( S p n ) ≤ ( 2 p − 2 ) p n − 2 $(2p-3)p^{n-2}\leq \gamma _{\mathrm {g}}(S_p^n), \gamma _{\mathrm {g}}^\prime (S_p^n) \leq (2p-2)p^{n-2}$ left parenthesis 2 p minus 3 right parenthesis p Superscript n minus 2 Baseline less than or equals gamma Subscript normal g Baseline left parenthesis upper S Subscript p Superscript n Baseline right parenthesis comma gamma prime Subscript normal g Baseline left parenthesis upper S Subscript p Superscript n Baseline right parenthesis less than or equals left parenthesis 2 p minus 2 right parenthesis p Superscript n minus 2
. For the total domination game, the bounds
γ tg ( S p n ) , γ tg ′ ( S p n ) ≥ ( 2 p − 2 ) p n − 2 $\gamma _{\mathrm {{tg}}}(S_p^n), \gamma _{\mathrm {{tg}}}^\prime (S_p^n) \geq (2p-2)p^{n-2}$ gamma Subscript tg Baseline left parenthesis upper S Subscript p Superscript n Baseline right parenthesis comma gamma prime Subscript tg Baseline left parenthesis upper S Subscript p Superscript n Baseline right parenthesis greater than or equals left parenthesis 2 p minus 2 right parenthesis p Superscript n minus 2
are established.

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