Bipartite graphs with close domination and k-domination numbers

Let $k$ be a positive integer and let $G$ be a graph with vertex set $V(G)$. A subset $D \subseteq V(G)$ is a $k$-dominating set if every vertex outside $D$ is adjacent to at least $k$ vertices in $D$. The $k$-domination number $\gamma_k(G)$ is the minimum cardinality of a $k$-dominating set in $G$. For any graph $G$, we know that $\gamma_k(G) \geq \gamma(G)+k-2$ where $\Delta(G)\geq k\geq 2$ and this bound is sharp for every $k\geq 2$. In this paper, we characterize bipartite graphs satisfying the equality for $k\geq 3$ and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when $k=3$. We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general

The super-connectivity of Johnson graphs

For positive integers $n,k$ and $t$, the uniform subset graph $G(n, k, t)$ has all $k$-subsets of $\{1,2,\ldots, n\}$ as vertices and two $k$-subsets are joined by an edge if they intersect at exactly $t$ elements. The Johnson graph $J(n,k)$ corresponds to $G(n,k,k-1)$, that is, two vertices of $J(n,k)$ are adjacent if the intersection of the corresponding $k$-subsets has size $k-1$. A super vertex-cut of a connected graph is a set of vertices whose removal disconnects the graph without isolating a vertex and the super-connectivity is the size of a minimum super vertex-cut. In this work, we fully determine the super-connectivity of the family of Johnson graphs $J(n,k)$ for $n\geq k\geq 1$