Abstract
Let [Formula: see text] be a graph. A subset [Formula: see text] of [Formula: see text] is called a dominating set of [Formula: see text] if every vertex not in [Formula: see text] is adjacent to some vertex in [Formula: see text]. The domination number [Formula: see text] of [Formula: see text] is the minimum cardinality taken over all dominating sets of [Formula: see text]. The shadow graph of [Formula: see text], denoted [Formula: see text] is the graph constructed from [Formula: see text] by taking two copies of [Formula: see text] namely [Formula: see text] itself and [Formula: see text] and by joining each vertex [Formula: see text] in [Formula: see text] to the neighbors of the corresponding vertex [Formula: see text] in [Formula: see text]. In this paper, we obtain the upper and lower bounds for the sum of domination number of a graph and its shadow graph and characterize such extremal graphs.
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