# On Happy Colorings, Cuts, and Structural Parameterizations

## TL;DR

This paper explores the relationships and parameterizations of the Maximum Happy Vertices and Edges problems, establishing new reductions and answering open questions in the field of graph clustering and cut problems.

## Contribution

It introduces a reduction between Maximum Happy Vertices and Node Multiway Cut and investigates structural parameterizations, addressing open questions from prior research.

## Key findings

- Established a reduction between Maximum Happy Vertices and Node Multiway Cut.
- Analyzed structural and distance to triviality parameterizations.
- Answered open questions from four previous studies.

## Abstract

We study the Maximum Happy Vertices and Maximum Happy Edges problems. The former problem is a variant of clusterization, where some vertices have already been assigned to clusters. The second problem gives a natural generalization of Multiway Uncut, which is the complement of the classical Multiway Cut problem. Due to their fundamental role in theory and practice, clusterization and cut problems has always attracted a lot of attention. We establish a new connection between these two classes of problems by providing a reduction between Maximum Happy Vertices and Node Multiway Cut. Moreover, we study structural and distance to triviality parameterizations of Maximum Happy Vertices and Maximum Happy Edges. Obtained results in these directions answer questions explicitly asked in four works: Agrawal '17, Aravind et al. '16, Choudhari and Reddy '18, Misra and Reddy '17.