arXiv subjects
Publications and source records attributed to Vinicius Fernandes dos Santos.
A graph is locally irregular if adjacent vertices have distinct degrees. A graph G is decomposable if its edge set can be decomposed into locally irregular graphs, and its locally irregular chromatic index lir(G) is the least number of graphs in such a decomposition. We prove that lir(G) <= 18 for every decomposable graph G, improving the previous bound of 220.
Given a property (graph class) $Π$, a graph $G$, and an integer $k$, the \emph{$Π$-completion} problem consists in deciding whether we can turn $G$ into a graph with the property $Π$ by adding at most $k$ edges to $G$. The $Π$-completion problem is known to be NP-hard for general graphs when $Π$ is the property of being a proper interval graph (PIG). In this work, we study the PIG-completion problem %when $Π$ is the class of proper interval graphs (PIG) within different subclasses of chordal graphs. We show that the problem remains NP-complete even when restricted to split graphs. We then turn our attention to positive results and present polynomial time algorithms to solve the PIG-completion problem when the input is restricted to caterpillar and threshold graphs. We also present an efficient algorithm for the minimum co-bipartite-completion for quasi-threshold graphs, which provides a lower bound for the PIG-completion problem within this graph class.
For a fixed property (graph class) $Π$, given a graph G and an integer k, the $Π$-deletion problem consists in deciding if we can turn $G$ into a graph with the property $Π$ by deleting at most $k$ edges. The $Π$-deletion problem is known to be NP-hard for most of the well-studied graph classes, such as chordal, interval, bipartite, planar, comparability and permutation graphs, among others; even deletion to cacti is known to be NP-hard for general graphs. However, there is a notable exception: the deletion problem to trees is polynomial. Motivated by this fact, we study the deletion problem for some classes similar to trees, addressing in this way a knowledge gap in the literature. We prove that deletion to cacti is hard even when the input is a bipartite graph. On the positive side, we show that the problem becomes tractable when the input is chordal, and for the special case of quasi-threshold graphs we give a simpler and faster algorithm. In addition, we present sufficient structural conditions on the graph class $Π$ that imply the NP-hardness of the $Π$-deletion problem, and show that deletion from general graphs to some well-known subclasses of forests is NP-hard.