Perfect Forests in Graphs and Their Extensions
Gregory Gutin, Anders Yeo

TL;DR
This paper investigates perfect forests in graphs, providing polynomial algorithms for some cases, proving NP-hardness for others, and characterizing conditions for the existence of certain perfect forests.
Contribution
It introduces new complexity results and algorithms for finding perfect forests with specific properties in graphs.
Findings
Polynomial-time algorithm for minimum edge 0-perfect forests.
NP-hardness of maximum edge 0-perfect forests.
W[1]-hardness for certain edge count thresholds.
Abstract
Let be a graph on vertices. For and a connected graph , a spanning forest of is called an -perfect forest if every tree in is an induced subgraph of and exactly vertices of have even degree (including zero). A -perfect forest of is proper if it has no vertices of degree zero. Scott (2001) showed that every connected graph with even number of vertices contains a (proper) 0-perfect forest. We prove that one can find a 0-perfect forest with minimum number of edges in polynomial time, but it is NP-hard to obtain a 0-perfect forest with maximum number of edges. Moreover, we show that to decide whether has a 0-perfect forest with at least edges, where is the parameter, is W[1]-hard. We also prove that for a prescribed edge of it is NP-hard to obtain a 0-perfect forest containing but one can decide…
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems · Limits and Structures in Graph Theory
