On the P-intersection graphs of acts over semigroups: applications to connectivity and monophonic convexity with clique structure

In this paper, we present the $\mathrm{\Psi }$-intersection graph ${\mathrm{\Psi }}_{\mathcal{S}}(\mathcal{B})$ associated with an $S$-act over topological semigroups. We investigate how the internal structure of $S$-acts, particularly the presence and distribution of simple subacts, determines fundamental graph-theoretic properties of ${\mathrm{\Psi }}_{\mathcal{S}}(\mathcal{B})$. Characterizations of connectivity, bipartiteness, girth, and diameter are established in terms of decomposition properties and chain conditions on subacts. We further analyze the clique structure of ${\mathrm{\Psi }}_{\mathcal{S}}(\mathcal{B})$, showing that every clique contains at most one simple subact and describing maximal cliques through families of non-simple subacts intersecting via simple components. Domination-type parameters are also studied by introducing indecomposable controlling sets and deriving bounds for the control number of ${\mathrm{\Psi }}_{\mathcal{S}}(\mathcal{B})$. Finally, by using monophonic convexity and the $m$-topology on graphs, we examine induced path structures and their interaction with algebraic paths in $S$-acts, revealing connections between geodesic behavior and semigroup actions. These results provide a unified algebraic, topological, and graph-theoretic framework and open new combinatorial and metric approaches to the study of topological path semigroups.