For a fixed property (graph class) $\Pi$, given a graph $G$ and an integer
$k$, the $\Pi$-deletion problem consists in deciding if we can turn $G$ into a
graph with the property $\Pi$ by deleting at most $k$ edges of $G$. The
$\Pi$-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), with the notable exception of trees.
Motivated by this fact, in this work we study the deletion problem for some
classes close to trees. We obtain NP-hardness results for several classes of
sparse graphs, for which we prove that deletion is hard even when the input is
a bipartite graph. In addition, we give sufficient structural conditions for
the graph class $\Pi$ for NP-hardness. In the case of deletion to cactus, we
show that the problem becomes tractable when the input is chordal, and we give
polynomial-time algorithms for quasi-threshold graphs.Comment: 12 pages, no figure

Koch, Ivo

Pardal, Nina

Santos, Vinicius Fernandes dos

English

arXiv

For a fixed property (graph class) ${\Pi}$, given a graph G and an integer k,
the ${\Pi}$-deletion problem consists in deciding if we can turn $G$ into a
graph with the property ${\Pi}$ by deleting at most $k$ edges. The
${\Pi}$-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 ${\Pi}$ that imply
the NP-hardness of the ${\Pi}$-deletion problem, and show that deletion from
general graphs to some well-known subclasses of forests is NP-hard.Comment: 10 pages, no figure

arXiv.org e-Print Archive

Edge deletion to tree-like graph classes

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

Edge deletion to tree-like graph classes

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