87 research outputs found
Solving MaxSAT and #SAT on structured CNF formulas
In this paper we propose a structural parameter of CNF formulas and use it to
identify instances of weighted MaxSAT and #SAT that can be solved in polynomial
time. Given a CNF formula we say that a set of clauses is precisely satisfiable
if there is some complete assignment satisfying these clauses only. Let the
ps-value of the formula be the number of precisely satisfiable sets of clauses.
Applying the notion of branch decompositions to CNF formulas and using ps-value
as cut function, we define the ps-width of a formula. For a formula given with
a decomposition of polynomial ps-width we show dynamic programming algorithms
solving weighted MaxSAT and #SAT in polynomial time. Combining with results of
'Belmonte and Vatshelle, Graph classes with structured neighborhoods and
algorithmic applications, Theor. Comput. Sci. 511: 54-65 (2013)' we get
polynomial-time algorithms solving weighted MaxSAT and #SAT for some classes of
structured CNF formulas. For example, we get algorithms for
formulas of clauses and variables and size , if has a linear
ordering of the variables and clauses such that for any variable occurring
in clause , if appears before then any variable between them also
occurs in , and if appears before then occurs also in any clause
between them. Note that the class of incidence graphs of such formulas do not
have bounded clique-width
Obstruction characterization of co-TT graphs
Threshold tolerance graphs and their complement graphs ( known as co-TT
graphs) were introduced by Monma, Reed and Trotter[24]. Introducing the concept
of negative interval Hell et al.[19] defined signed-interval bigraphs/digraphs
and have shown that they are equivalent to several seemingly different classes
of bigraphs/digraphs. They have also shown that co-TT graphs are equivalent to
symmetric signed-interval digraphs. In this paper we characterize
signed-interval bigraphs and signed-interval graphs respectively in terms of
their biadjacency matrices and adjacency matrices. Finally, based on the
geometric representation of signed-interval graphs we have setteled the open
problem of forbidden induced subgraph characterization of co-TT graphs posed by
Monma, Reed and Trotter in the same paper.Comment: arXiv admin note: substantial text overlap with arXiv:2206.0591
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
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