Abstract Let $$ {\mathcal {H}} $$ H be a connected subgraph of a graph G. The $${\mathcal {H}}$$ H -structure connectivity of G, denoted by $$ \kappa (G;{\mathcal {H}}) $$ κ ( G ; H ) , is the minimum cardinality of a set of connected subgraphs in G, whose removal either disconnects G or reduces it to a trivial graph, where each element in the set is isomorphic to $$ {\mathcal {H}} $$ H . The $${\mathcal {H}}$$ H -substructure connectivity of G, denoted by $$ \kappa ^s(G;{\mathcal {H}}) $$ κ s ( G ; H ) , is the minimum cardinality of a set of connected subgraphs in G, whose removal either disconnects G or reduces it to a trivial graph, where each element in the set is isomorphic to a connected subgraph of $$ {\mathcal {H}} $$ H . In this paper, we investigate the $$ {\mathcal {H}} $$ H -structure connectivity and $$ {\mathcal {H}} $$ H -substructure connectivity of folded divide-and-swap cube $$ FDSC_n $$ F D S C n for $$ {\mathcal {H}}\in \{K_1, K_{1,1}, K_{1,m} \text (2\le m \le d+2) \} $$ H ∈ { K 1 , K 1 , 1 , K 1 , m ( 2 ≤ m ≤ d + 2 ) } where $$ n=2^d $$ n = 2 d . We show that $$\kappa (FDSC_n;K_1)=\kappa ^s(FDSC_n;K_1)=d+2$$ κ ( F D S C n ; K 1 ) = κ s ( F D S C n ; K 1 ) = d + 2 , $$\kappa (FDSC_n;K_{1,1})=\kappa ^s(FDSC_n;K_{1,1})=d+1 $$ κ ( F D

Abstract Let $ G $ be a graph. A set $ S $ of vertices in $ G $ is a disjunctive total dominating set of $ G $ if every vertex is adjacent to a vertex of $ S $ or has at least two vertices in $ S $ at distance two from it. The disjunctive total domination number, $ \gamma _t^d(G) $, is the minimum cardinality of such a set. In this paper, we determine disjunctive total domination number of Harary graph $ H_{k,n} $ for all $ k $ and $ n $.

A set S ⊆ V (G) is a disjunctive total dominating set of G if every vertex has a neighbor in S or has at least two vertices in S at distance 2 from it. The disjunctive total domination number is the minimum cardinality of a disjunctive total dominating set in G. We define the disjunctive total domi nation subdivision number of G as the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) to increase the disjunctive total domination number. In this paper, we first study the disjunctive total domination subdivision number of some special graphs. Next, we give some upper bounds on the disjunctive total domination subdivision number for any graphs in terms of vertex degree. Finally, we supply some conditions for a graph G to have a minimum disjunctive total domination subdivision number.