Let G = (V (G), E(G)) be an undirected connected graph and let
X be a subset of V (G). The set I(X) = XN(X) denotes the set of
isolates in X and B(X) = (V (G) X) ∩ N(X) denotes the boundary
of X, where N(X) = {y ∈ V (G) : yx ∈ E(G) for some x ∈ X} is
the set of neighbors of X. The I-differential of X is given by ∂I (X) =
|B(X)| − |I(X)|. The I-differential of G, denoted by ∂I (G), is equal
to max{∂I (X) : X is a subset of V (G)}. In [3], Lewis et al. showed
that ∂I (G) ≤ n − γt(G), where n is the order of G and γt(G) is the
total domination number of G. Then, in [5], Caga-anan and Canoy
showed that this is actually an equality, thereby showing that finding
the I-differential of a graph is equivalent to finding its total domination
number. In this paper, we introduce the I-integral of a graph and
present some important results, some of which showing the relationship
of the I-integral, I-differential, total domination number, and domination
number of a graph.
Let G = (V (G), E(G)) be an undirected connected graph and letX be a subset of V (G). The set I(X) = XN(X) denotes the set ofisolates in X and B(X) = (V (G) X) ∩ N(X) denotes the boundaryof X, where N(X) = {y ∈ V (G) : yx ∈ E(G) for some x ∈ X} isthe set of neighbors of X. The I-differential of X is given by ∂I (X) =|B(X)| − |I(X)|. The I-differential of G, denoted by ∂I (G), is equalto max{∂I (X) : X is a subset of V (G)}. In [3], Lewis et al. showedthat ∂I (G) ≤ n − γt(G), where n is the order of G and γt(G) is thetotal domination number of G. Then, in [5], Caga-anan and Canoyshowed that this is actually an equality, thereby showing that findingthe I-differential of a graph is equivalent to finding its total dominationnumber. In this paper, we introduce the I-integral of a graph andpresent some important results, some of which showing the relationshipof the I-integral, I-differential, total domination number, and dominationnumber of a graph.
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