11 research outputs found

### Algorithmic Meta-Theorems for Combinatorial Reconfiguration Revisited

Given a graph and two vertex sets satisfying a certain feasibility condition,
a reconfiguration problem asks whether we can reach one vertex set from the
other by repeating prescribed modification steps while maintaining feasibility.
In this setting, Mouawad et al. [IPEC 2014] presented an algorithmic
meta-theorem for reconfiguration problems that says if the feasibility can be
expressed in monadic second-order logic (MSO), then the problem is
fixed-parameter tractable parameterized by $\textrm{treewidth} + \ell$, where
$\ell$ is the number of steps allowed to reach the target set. On the other
hand, it is shown by Wrochna [J. Comput. Syst. Sci. 2018] that if $\ell$ is not
part of the parameter, then the problem is PSPACE-complete even on graphs of
bounded bandwidth.
In this paper, we present the first algorithmic meta-theorems for the case
where $\ell$ is not part of the parameter, using some structural graph
parameters incomparable with bandwidth. We show that if the feasibility is
defined in MSO, then the reconfiguration problem under the so-called token
jumping rule is fixed-parameter tractable parameterized by neighborhood
diversity. We also show that the problem is fixed-parameter tractable
parameterized by $\textrm{treedepth} + k$, where $k$ is the size of sets being
transformed. We finally complement the positive result for treedepth by showing
that the problem is PSPACE-complete on forests of depth $3$.Comment: 25 pages, 2 figures, ESA 202

### Algorithmic Meta-Theorems for Combinatorial Reconfiguration Revisited

Given a graph and two vertex sets satisfying a certain feasibility condition, a reconfiguration problem asks whether we can reach one vertex set from the other by repeating prescribed modification steps while maintaining feasibility. In this setting, Mouawad et al. [IPEC 2014] presented an algorithmic meta-theorem for reconfiguration problems that says if the feasibility can be expressed in monadic second-order logic (MSO), then the problem is fixed-parameter tractable parameterized by treewidth + ?, where ? is the number of steps allowed to reach the target set. On the other hand, it is shown by Wrochna [J. Comput. Syst. Sci. 2018] that if ? is not part of the parameter, then the problem is PSPACE-complete even on graphs of bounded bandwidth.
In this paper, we present the first algorithmic meta-theorems for the case where ? is not part of the parameter, using some structural graph parameters incomparable with bandwidth. We show that if the feasibility is defined in MSO, then the reconfiguration problem under the so-called token jumping rule is fixed-parameter tractable parameterized by neighborhood diversity. We also show that the problem is fixed-parameter tractable parameterized by treedepth + k, where k is the size of sets being transformed. We finally complement the positive result for treedepth by showing that the problem is PSPACE-complete on forests of depth 3

### Orientable Burning Number of Graphs

In this paper, we introduce the problem of finding an orientation of a given
undirected graph that maximizes the burning number of the resulting directed
graph. We show that the problem is polynomial-time solvable on
K\H{o}nig-Egerv\'{a}ry graphs (and thus on bipartite graphs) and that an almost
optimal solution can be computed in polynomial time for perfect graphs. On the
other hand, we show that the problem is NP-hard in general and W[1]-hard
parameterized by the target burning number. The hardness results are
complemented by several fixed-parameter tractable results parameterized by
structural parameters. Our main result in this direction shows that the problem
is fixed-parameter tractable parameterized by cluster vertex deletion number
plus clique number (and thus also by vertex cover number).Comment: 17pages, 3 figures, WALCOM 202

### Structural Parameterizations of Vertex Integrity

The graph parameter vertex integrity measures how vulnerable a graph is to a
removal of a small number of vertices. More precisely, a graph with small
vertex integrity admits a small number of vertex removals to make the remaining
connected components small. In this paper, we initiate a systematic study of
structural parameterizations of the problem of computing the
unweighted/weighted vertex integrity. As structural graph parameters, we
consider well-known parameters such as clique-width, treewidth, pathwidth,
treedepth, modular-width, neighborhood diversity, twin cover number, and
cluster vertex deletion number. We show several positive and negative results
and present sharp complexity contrasts.Comment: 21 pages, 6 figures, WALCOM 202

### Extended MSO Model Checking via Small Vertex Integrity

We study the model checking problem of an extended $\mathsf{MSO}$ with local
and global cardinality constraints, called
$\mathsf{MSO}^{\mathsf{GL}}_{\mathsf{Lin}}$, introduced recently by Knop,
Kouteck\'{y}, Masa\v{r}\'{i}k, and Toufar [Log. Methods Comput. Sci., 15(4),
2019]. We show that the problem is fixed-parameter tractable parameterized by
vertex integrity, where vertex integrity is a graph parameter standing between
vertex cover number and treedepth. Our result thus narrows the gap between the
fixed-parameter tractability parameterized by vertex cover number and the
W[1]-hardness parameterized by treedepth

### Exploring the Gap Between Treedepth and Vertex Cover Through Vertex Integrity

For intractable problems on graphs of bounded treewidth, two graph parameters
treedepth and vertex cover number have been used to obtain fine-grained
complexity results. Although the studies in this direction are successful, we
still need a systematic way for further investigations because the graphs of
bounded vertex cover number form a rather small subclass of the graphs of
bounded treedepth. To fill this gap, we use vertex integrity, which is placed
between the two parameters mentioned above. For several graph problems, we
generalize fixed-parameter tractability results parameterized by vertex cover
number to the ones parameterized by vertex integrity. We also show some finer
complexity contrasts by showing hardness with respect to vertex integrity or
treedepth.Comment: 30 pages, 5 figures, CIAC 202