43 research outputs found
Graph isomorphism completeness for trapezoid graphs
The complexity of the graph isomorphism problem for trapezoid graphs has been
open over a decade. This paper shows that the problem is GI-complete. More
precisely, we show that the graph isomorphism problem is GI-complete for
comparability graphs of partially ordered sets with interval dimension 2 and
height 3. In contrast, the problem is known to be solvable in polynomial time
for comparability graphs of partially ordered sets with interval dimension at
most 2 and height at most 2.Comment: 4 pages, 3 Postscript figure
Graph classes equivalent to 12-representable graphs
Jones et al. (2015) introduced the notion of -representable graphs, where
is a word over different from , as a generalization
of word-representable graphs. Kitaev (2016) showed that if is of length at
least 3, then every graph is -representable. This indicates that there are
only two nontrivial classes in the theory of -representable graphs:
11-representable graphs, which correspond to word-representable graphs, and
12-representable graphs. This study deals with 12-representable graphs.
Jones et al. (2015) provided a characterization of 12-representable trees in
terms of forbidden induced subgraphs. Chen and Kitaev (2022) presented a
forbidden induced subgraph characterization of a subclass of 12-representable
grid graphs.
This paper shows that a bipartite graph is 12-representable if and only if it
is an interval containment bigraph. The equivalence gives us a forbidden
induced subgraph characterization of 12-representable bipartite graphs since
the list of minimal forbidden induced subgraphs is known for interval
containment bigraphs. We then have a forbidden induced subgraph
characterization for grid graphs, which solves an open problem of Chen and
Kitaev (2022). The study also shows that a graph is 12-representable if and
only if it is the complement of a simple-triangle graph. This equivalence
indicates that a necessary condition for 12-representability presented by Jones
et al. (2015) is also sufficient. Finally, we show from these equivalences that
12-representability can be determined in time for bipartite graphs and
in time for arbitrary graphs, where and are the
number of vertices and edges of the complement of the given graph.Comment: 12 pages, 6 figure
A recognition algorithm for adjusted interval digraphs
Min orderings give a vertex ordering characterization, common to some graphs
and digraphs such as interval graphs, complements of threshold tolerance graphs
(known as co-TT graphs), and two-directional orthogonal ray graphs. An adjusted
interval digraph is a reflexive digraph that has a min ordering. Adjusted
interval digraph can be recognized in time, where is the number of
vertices of the given graph. Finding a more efficient algorithm is posed as an
open question. This note provides a new recognition algorithm with running time
. The algorithm produces a min ordering if the given graph is an
adjusted interval digraph
Complexity of Hamiltonian Cycle Reconfiguration
The Hamiltonian cycle reconfiguration problem asks, given two Hamiltonian cycles C 0 and C t of a graph G, whether there is a sequence of Hamiltonian cycles C 0 , C 1 , … , C t such that C i can be obtained from C i − 1 by a switch for each i with 1 ≤ i ≤ t , where a switch is the replacement of a pair of edges u v and w z on a Hamiltonian cycle with the edges u w and v z of G, given that u w and v z did not appear on the cycle. We show that the Hamiltonian cycle reconfiguration problem is PSPACE-complete, settling an open question posed by Ito et al. (2011) and van den Heuvel (2013). More precisely, we show that the Hamiltonian cycle reconfiguration problem is PSPACE-complete for chordal bipartite graphs, strongly chordal split graphs, and bipartite graphs with maximum degree 6. Bipartite permutation graphs form a proper subclass of chordal bipartite graphs, and unit interval graphs form a proper subclass of strongly chordal graphs. On the positive side, we show that, for any two Hamiltonian cycles of a bipartite permutation graph and a unit interval graph, there is a sequence of switches transforming one cycle to the other, and such a sequence can be obtained in linear time