The inclusion ideal graph In(S)of a semigroup S is an undirected simple graph whose vertices are all the nontrivial left ideals of S and two distinct left ideals I,J are adjacent if and only if either I ⊂ J or J ⊂ I.The purpose of this paper is to study algebraic properties of the semigroup S as well as graph theoretic properties of In(S).We investigate the connectedness of In(S)and show that the diameter of In(S)is at most 3 if it is connected.We also obtain a necessary and sufficient condition of S such that the clique number of In(S)is the number of minimal left ideals of S.Further,various graph invariants of In(S),viz.perfectness,planarity,girth,etc.,are discussed.For a completely simple semigroup S,we investigate properties of In(S)including its independence number and matching number.Finally,we obtain the automorphism group of In(S).
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