A graph is said to be an intersection graph if there is a set of objects such that each vertex corresponds to an object and two vertices are adjacent if and only if the corresponding objects have a nonempty intersection. There are several natural graph classes that have geometric intersection representations. The geometric representations sometimes help to prove tractability/intractability of p...

‎suppose $n$ is a fixed positive integer‎. ‎we introduce the relative n-th non-commuting graph $gamma^{n} _{h,g}$‎, ‎associated to the non-abelian subgroup $h$ of group $g$‎. ‎the vertex set is $gsetminus c^n_{h,g}$ in which $c^n_{h,g} = {xin g‎ : ‎[x,y^{n}]=1 mbox{~and~} [x^{n},y]=1mbox{~for~all~} yin h}$‎. ‎moreover‎, ‎${x,y}$ is an edge if $x$ or $y$ belong to $h$ and $xy^{n}eq y^{n}x$ or $x...

The intersection number of a graph G is the minimum size of a ground set S such that G is an intersection graph of some family of subsets F ⊆ 2 . The overlap number of G is defined similarly, except that G is required to be an overlap graph of F . In this paper we show two algorithmic aspects concerning both these graph invariants. On the one hand, we show that the corresponding optimization pr...

A string graph is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of almost all string graphs on n vertices can be partitioned into five cliques such that so...

The intersection ideal graph Γ(S) of a semigroup S is simple undirected whose vertices are all nontrivial left ideals and two distinct I, J adjacent if only their nontrivial. In this paper, we investigate the connectedness Γ(S). We show that connected, then diameter at most two. Further, classify semigroups in terms such obtain domination number, independence girth strong metric dimension have ...