In a vertex-colored graph, an edge is happy if its endpoints have the same color. Similarly, a vertex is happy if all its incident edges are happy. Motivated by the computation of homophily in social networks, we consider the algorithmic aspects of the following Maximum Happy Edges ( k-MHE ) problem: given a partially k-colored graph G and an integer ℓ, find an extended full k-coloring of G making at least ℓ edges happy. When we want to make ℓ vertices happy on the same input, the problem is known as Maximum Happy Vertices ( k-MHV ). We perform an extensive study into the complexity of the problems, particularly from a parameterized viewpoint. For every $k\ge 3$, we prove both problems can be solved in time $2^n{n}^{O(1)}$. Moreover, by combining this result with a linear vertex kernel of size $(k+\ell )$ for k-MHE, we show that the edge-variant can be solved in time $2^{\ell }{n}^{O(1)}$. In contrast, we prove that the vertex-variant remains W[1]-hard for the natural parameter ℓ. However, the problem does admit a kernel with $O(k^2{\ell }^2)$ vertices for the combined parameter $k+\ell$. From a structural perspective, we show both problems are fixed-parameter tractable for treewidth and neighborhood diversity, which can both be seen as sparsity and density measures of a graph. Finally, we extend the known

在顶点彩色图形中，如果边缘的端点具有相同的颜色，则该边缘是快乐的。同样，一个顶点是幸福的，如果它的所有关联边是幸福的。基于社交网络中同构性的计算，我们考虑了以下最大快乐边缘（k -MHE）问题的算法方面：给定部分k色图G和整数ℓ，找到G的扩展全k色使至少ℓ个边缘开心。当我们想在同一输入上使ℓ个顶点快乐时，该问题称为最大快乐顶点（k -MHV）。我们对问题的复杂性进行了广泛的研究，尤其是从参数化的角度来看。对于每个$ķ\ge 3$，我们证明两个问题都可以及时解决 $2^{ñ}{ñ}^{Ø（\mathrm{1个}）}$。此外，通过将此结果与大小为1的线性顶点核相结合$（ķ+\ell ）$对于k -MHE，我们表明可以及时解决边缘变化$2^{\ell }{ñ}^{Ø（\mathrm{1个}）}$。相反，我们证明对于自然参数ℓ，顶点变量保持W [1] -hard 。但是，该问题确实允许使用$Ø（{ķ}^2{\ell }^2）$ 组合参数的顶点 $ķ+\ell$。从结构的角度来看，我们显示这两个问题对于树宽和邻域分集都是固定参数可处理的，这两个问题都可以视为图的稀疏性和密度度量。最后，我们扩展已知的