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We study a new variant of graph coloring by adding a connectivity constraint. A path in a vertex-colored graph is called conflict-free if there is a color that appears exactly once on its vertices. A connected graph $G$ is said to be strongly conflict-free vertex-connection $k$-colorable if $G$ admits a vertex $k$-coloring such that any two distinct vertices of $G$ are connected by a conflict-free $shortest$ path. Among others, we show that deciding whether a given graph is strongly conflict-free vertex-connection $3$-colorable is NP-complete even when restricted to $3$-colorable graphs with diameter $3$, radius $2$ and domination number $3$, and, assuming the Exponential Time Hypothesis (ETH), cannot be solved in $2^{o(n)}$ time on such restricted input graphs with $n$ vertices. This hardness result is quite strong when compared to the ordinary $3$-COLORING problem: it is known that $3$-COLORING is solvable in polynomial time in graphs with bounded domination number, and assuming ETH, cannot be solved in $2^{o(\sqrt{n})}$ time in $n$-vertex graphs with diameter $3$ and radius $2$. On the positive side, we point out that a strong conflict-free vertex-connection coloring with minimum color number of a given split graph or a co-bipartite graph can be computed in polynomial time., Comment: The full version of a COCOON 2024 paper

Let $d$-claw (or $d$-star) stand for $K_{1,d}$, the complete bipartite graph with 1 and $d\ge 1$ vertices on each part. The $d$-claw vertex deletion problem, $d$-CLAW-VD, asks for a given graph $G$ and an integer $k$ if one can delete at most $k$ vertices from $G$ such that the resulting graph has no $d$-claw as an induced subgraph. Thus, 1-CLAW-VD and 2-CLAW-VD are just the famous VERTEX COVER problem and the CLUSTER VERTEX DELETION problem, respectively. In this paper, we strengthen a hardness result in [M. Yannakakis, Node-Deletion Problems on Bipartite Graphs, SIAM J. Comput. (1981)], by showing that CLUSTER VERTEX DELETION remains NP-complete when restricted to bipartite graphs of maximum degree 3. Moreover, for every $d\ge 3$, we show that $d$-CLAW-VD is NP-complete even when restricted to bipartite graphs of maximum degree $d$. These hardness results are optimal with respect to degree constraint. By extending the hardness result in [F. Bonomo-Braberman et al., Linear-Time Algorithms for Eliminating Claws in Graphs, COCOON 2020], we show that, for every $d\ge 3$, $d$-CLAW-VD is NP-complete even when restricted to split graphs without $(d+1)$-claws, and split graphs of diameter 2. On the positive side, we prove that $d$-CLAW-VD is polynomially solvable on what we call $d$-block graphs, a class properly contains all block graphs. This result extends the polynomial-time algorithm in [Y. Cao et al., Vertex deletion problems on chordal graphs, Theor. Comput. Sci. (2018)] for 2-CLAW-VD on block graphs to $d$-CLAW-VD for all $d\ge 2$ and improves the polynomial-time algorithm proposed by F. Bonomo-Brabeman et al. for (unweighted) 3-CLAW-VD on block graphs to 3-block graphs.