A vertex-coloring of a connected graph $G$ is a strong conflict-free vertex-connection coloring if every two distinct vertices are joined by a shortest path on which some color appears exactly once. The minimum number of colors in such a coloring is the strong conflict-free vertex-connection number $\operatorname{svcfc}(G)$. We study this problem under the parameter twin cover. Let $X$ be a twin cover of $G$ of size $t$, and let $k$ be the target number of colors. In our first result, given $(G,k)$ together with a twin cover $X$, we reduce in polynomial time to an equivalent annotated instance on at most $\max\{2,t+(t+1)k2^{t+k-1}\}$ vertices. Hence the annotated version of Strong CFVC Number, in which a twin cover is supplied as part of the input, is fixed-parameter tractable parameterized by $t+k$. Using this bound, we then obtain a kernel parameterized by $\operatorname{tc}(G)+k$; in particular, for every fixed $k$, the problem is fixed-parameter tractable parameterized by the twin-cover number alone. In our second result, we prove every connected graph $G$ with twin cover $X$ of size $t$ satisfies $χ(G)\le \operatorname{svcfc}(G)\le χ(G)+t$. More generally, if $Y\subseteq X$ intersects every shortest path of length at least $3$, then $\operatorname{svcfc}(G)\le χ(G)+|Y|$. We also derive an exact expression for the chromatic number on graphs of bounded twin-cover number: for every proper coloring $\varphi$ of $G[X]$, the minimum number of colors needed to extend $\varphi$ to all of $G$ is $K_\varphi=\max_{S\subseteq X}(|\varphi(S)|+m(S))$, and hence $χ(G)=\min_{\varphi\text{ proper on }G[X]} K_\varphi$. Our results provide the first evidence that twin cover is a useful parameter for strong conflict-free vertex-connection and show that, once a twin cover is fixed, the remaining difficulty is concentrated in a bounded additive gap above the chromatic number.

翻译：连通图$G$的一个顶点着色称为强无冲突顶点连接着色，若每一对不同的顶点之间存在一条最短路径，在该路径上某种颜色恰好出现一次。这种着色所需的最小颜色数称为强无冲突顶点连接数$\operatorname{svcfc}(G)$。我们以孪生覆盖为参数研究该问题。设$X$是$G$的一个大小为$t$的孪生覆盖，$k$为目标颜色数。首先，给定$(G,k)$及孪生覆盖$X$，我们在多项式时间内将其归约为一个至多包含$\max\{2,t+(t+1)k2^{t+k-1}\}$个顶点的等价的带标注实例。因此，强CFVC数（Strong CFVC Number）的带标注版本（输入中附带孪生覆盖）在以$t+k$为参数时是固定参数可解的。利用这一界，我们进一步得到一个以$\operatorname{tc}(G)+k$为参数的核；特别地，对每个固定的$k$，该问题在以孪生覆盖数为参数时是固定参数可解的。其次，我们证明：若连通图$G$的孪生覆盖$X$的大小为$t$，则$χ(G)\le \operatorname{svcfc}(G)\le χ(G)+t$。更一般地，若$Y\subseteq X$与所有长度至少为$3$的最短路径相交，则$\operatorname{svcfc}(G)\le χ(G)+|Y|$。我们还推导了具有有界孪生覆盖数的图的色数的精确表达式：对$G[X]$的每一个正常着色$\varphi$，将$\varphi$延拓至整个$G$所需的最小颜色数为$K_\varphi=\max_{S\subseteq X}(|\varphi(S)|+m(S))$，从而$χ(G)=\min_{\varphi\text{ 在 }G[X]\text{上正常}} K_\varphi$。我们的结果首次证明孪生覆盖是强无冲突顶点连接的有效参数，并表明一旦固定孪生覆盖，剩余难度集中在色数之上的一个有界加性间隙中。