On Exponential Domination of Cm × Cn

نویسندگان

  • Mark Anderson
  • Robert C. Brigham
  • Julie R. Carrington
  • Richard P. Vitray
  • Jay Yellen
چکیده

An exponential dominating set of graph G = (V,E) is a subset D ⊆ V such that ∑ w∈D( 1 2 )d(v,w)−1 ≥ 1 for every v ∈ V, where d(v, w) is the distance between vertices v and w. The exponential domination number, γe(G), is the smallest cardinality of an exponential dominating set. Lower and upper bounds for γe(Cm × Cn) are determined and it is shown that limm,n→∞ γe(Cm×Cn) mn ≤ 1 13 . Two connections are also established between exponential domination and distance-2 domination: (a) If D is an exponential dominating set of the infinite grid graph such that no two vertices in D are closer than distance 5, then D is a distance-2 dominating set; and (b) For sufficiently large m and n, every distance-2 dominating set of Cm ×Cn is an exponential dominating set.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Lower Bounds for the Exponential Domination Number of Cm × Cn

A vertex v in an exponential dominating set assigns weight ( 1 2 )dist(v,u) to vertex u. An exponential dominating set of a graph G is a subset of V (G) such that every vertex in V (G) has been assigned a sum weight of at least 1. In this paper the exponential dominating number for the graph Cm × Cn, denoted by γe(Cm × Cn) is discussed. Anderson et. al. [1] proved that mn 15.875 ≤ γe(Cm × Cn) ≤...

متن کامل

On exponential domination and graph operations

An exponential dominating set of graph $G = (V,E )$ is a subset $Ssubseteq V(G)$ such that $sum_{uin S}(1/2)^{overline{d}{(u,v)-1}}geq 1$ for every vertex $v$ in $V(G)-S$, where $overline{d}(u,v)$ is the distance between vertices $u in S$ and $v  in V(G)-S$ in the graph $G -(S-{u})$. The exponential domination number, $gamma_{e}(G)$, is the smallest cardinality of an exponential dominating set....

متن کامل

Split Domination in Normal Product of Paths and Cycles

A dominating set D ⊆ V is a split dominating set of a graph G = (V,E) if the induced subgraph of 〈V −D〉 is disconnected. The split domination number γs(G) is the minimum cardinality of a split dominating set of a graph G. In this article, we establish some results on split domination number of Pm⊕ Pn, Pm ⊕ Cn and Cm ⊕ Cn.

متن کامل

On the domination of Cartesian product of directed cycles: Results for certain equivalence classes of 2 lengths

Let γ( −→ Cm2 −→ Cn) be the domination number of the Cartesian product of directed 6 cycles −→ Cm and −→ Cn for m,n ≥ 2. Shaheen [13] and Liu et al.([11], [12]) determined the value of γ( −→ Cm2 −→ Cn) when m ≤ 6 and [12] when both m and n ≡ 0 (mod 3). In 8 this article we give, in general, the value of γ( −→ Cm2 −→ Cn) when m ≡ 2 (mod 3) and improve the known lower bounds for most of the remai...

متن کامل

On the Domination of Cartesian Product of Directed Cycles: Results for Certain Equivalence Classes of Lengths

Let γ( −→ Cm2 −→ Cn) be the domination number of the Cartesian product of directed cycles −→ Cm and −→ Cn for m,n ≥ 2. Shaheen [13] and Liu et al. ([11], [12]) determined the value of γ( −→ Cm2 −→ Cn) when m ≤ 6 and [12] when both m and n ≡ 0(mod 3). In this article we give, in general, the value of γ(−→ Cm2 −→ Cn) when m ≡ 2(mod 3) and improve the known lower bounds for most of the remaining c...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2009