10 research outputs found
TS-Reconfiguration of Dominating Sets in circle and circular-arc graphs
We study the dominating set reconfiguration problem with the token sliding
rule. It consists, given a graph G=(V,E) and two dominating sets D_s and D_t of
G, in determining if there exists a sequence S= of
dominating sets of G such that for any two consecutive dominating sets D_r and
D_{r+1} with r<t, D_{r+1}=(D_r\ u) U v, where uv is an edge of G.
In a recent paper, Bonamy et al studied this problem and raised the following
questions: what is the complexity of this problem on circular arc graphs? On
circle graphs? In this paper, we answer both questions by proving that the
problem is polynomial on circular-arc graphs and PSPACE-complete on circle
graphs.Comment: This work was supported by ANR project GrR (ANR-18-CE40-0032) and
submitted to the conference WADS 202
Shortest Reconfiguration of Matchings
Imagine that unlabelled tokens are placed on the edges of a graph, such that
no two tokens are placed on incident edges. A token can jump to another edge if
the edges having tokens remain independent. We study the problem of determining
the distance between two token configurations (resp., the corresponding
matchings), which is given by the length of a shortest transformation. We give
a polynomial-time algorithm for the case that at least one of the two
configurations is not inclusion-wise maximal and show that otherwise, the
problem admits no polynomial-time sublogarithmic-factor approximation unless P
= NP. Furthermore, we show that the distance of two configurations in bipartite
graphs is fixed-parameter tractable parameterized by the size of the
symmetric difference of the source and target configurations, and obtain a
-factor approximation algorithm for every if
additionally the configurations correspond to maximum matchings. Our two main
technical tools are the Edmonds-Gallai decomposition and a close relation to
the Directed Steiner Tree problem. Using the former, we also characterize those
graphs whose corresponding configuration graphs are connected. Finally, we show
that deciding if the distance between two configurations is equal to a given
number is complete for the class , and deciding if the diameter of
the graph of configurations is equal to is -hard.Comment: 31 pages, 3 figure
Parameterized Complexities of Dominating and Independent Set Reconfiguration
We settle the parameterized complexities of several variants of independent
set reconfiguration and dominating set reconfiguration, parameterized by the
number of tokens. We show that both problems are XL-complete when there is no
limit on the number of moves and XNL-complete when a maximum length for
the sequence is given in binary in the input. The problems are known to be
XNLP-complete when is given in unary instead, and - and
-hard respectively when is also a parameter. We complete the
picture by showing membership in those classes.
Moreover, we show that for all the variants that we consider, token sliding
and token jumping are equivalent under pl-reductions. We introduce partitioned
variants of token jumping and token sliding, and give pl-reductions between the
four variants that have precise control over the number of tokens and the
length of the reconfiguration sequence.Comment: 31 pages, 3 figure