8,146 research outputs found
Partial Complementation of Graphs
A partial complement of the graph G is a graph obtained from G by complementing all the edges in one of its induced subgraphs. We study the following algorithmic question: for a given graph G and graph class G, is there a partial complement of G which is in G? We show that this problem can be solved in polynomial time for various choices of the graphs class G, such as bipartite, degenerate, or cographs. We complement these results by proving that the problem is NP-complete when G is the class of r-regular graphs
Hypomorphy of graphs up to complementation
Let be a set of cardinality (possibly infinite). Two graphs and
with vertex set are {\it isomorphic up to complementation} if is
isomorphic to or to the complement of . Let be a
non-negative integer, and are {\it -hypomorphic up to
complementation} if for every -element subset of , the induced
subgraphs and are isomorphic up to
complementation. A graph is {\it -reconstructible up to complementation}
if every graph which is -hypomorphic to up to complementation is in
fact isomorphic to up to complementation. We give a partial
characterisation of the set of pairs such that two graphs
and on the same set of vertices are equal up to complementation
whenever they are -hypomorphic up to complementation. We prove in particular
that contains all pairs such that . We
also prove that 4 is the least integer such that every graph having a
large number of vertices is -reconstructible up to complementation; this
answers a question raised by P. Ill
The Group Structure of Pivot and Loop Complementation on Graphs and Set Systems
We study the interplay between principal pivot transform (pivot) and loop
complementation for graphs. This is done by generalizing loop complementation
(in addition to pivot) to set systems. We show that the operations together,
when restricted to single vertices, form the permutation group S_3. This leads,
e.g., to a normal form for sequences of pivots and loop complementation on
graphs. The results have consequences for the operations of local
complementation and edge complementation on simple graphs: an alternative proof
of a classic result involving local and edge complementation is obtained, and
the effect of sequences of local complementations on simple graphs is
characterized.Comment: 21 pages, 7 figures, significant additions w.r.t. v3 are Thm 7 and
Remark 2
On the Classification of All Self-Dual Additive Codes over GF(4) of Length up to 12
We consider additive codes over GF(4) that are self-dual with respect to the
Hermitian trace inner product. Such codes have a well-known interpretation as
quantum codes and correspond to isotropic systems. It has also been shown that
these codes can be represented as graphs, and that two codes are equivalent if
and only if the corresponding graphs are equivalent with respect to local
complementation and graph isomorphism. We use these facts to classify all codes
of length up to 12, where previously only all codes of length up to 9 were
known. We also classify all extremal Type II codes of length 14. Finally, we
find that the smallest Type I and Type II codes with trivial automorphism group
have length 9 and 12, respectively.Comment: 18 pages, 4 figure
Boundedness in languages of infinite words
We define a new class of languages of -words, strictly extending
-regular languages.
One way to present this new class is by a type of regular expressions. The
new expressions are an extension of -regular expressions where two new
variants of the Kleene star are added: and . These new
exponents are used to say that parts of the input word have bounded size, and
that parts of the input can have arbitrarily large sizes, respectively. For
instance, the expression represents the language of infinite
words over the letters where there is a common bound on the number of
consecutive letters . The expression represents a similar
language, but this time the distance between consecutive 's is required to
tend toward the infinite.
We develop a theory for these languages, with a focus on decidability and
closure. We define an equivalent automaton model, extending B\"uchi automata.
The main technical result is a complementation lemma that works for languages
where only one type of exponent---either or ---is used.
We use the closure and decidability results to obtain partial decidability
results for the logic MSOLB, a logic obtained by extending monadic second-order
logic with new quantifiers that speak about the size of sets
Ribbon graphs and bialgebra of Lagrangian subspaces
To each ribbon graph we assign a so-called L-space, which is a Lagrangian
subspace in an even-dimensional vector space with the standard symplectic form.
This invariant generalizes the notion of the intersection matrix of a chord
diagram. Moreover, the actions of Morse perestroikas (or taking a partial dual)
and Vassiliev moves on ribbon graphs are reinterpreted nicely in the language
of L-spaces, becoming changes of bases in this vector space. Finally, we define
a bialgebra structure on the span of L-spaces, which is analogous to the
4-bialgebra structure on chord diagrams.Comment: 21 pages, 13 figures. v2: major revision, Sec 2 and 3 completely
rewritten; v3: minor corrections. Final version, to appear in Journal of Knot
Theory and its Ramification
Finding Large H-Colorable Subgraphs in Hereditary Graph Classes
We study the \textsc{Max Partial -Coloring} problem: given a graph ,
find the largest induced subgraph of that admits a homomorphism into ,
where is a fixed pattern graph without loops. Note that when is a
complete graph on vertices, the problem reduces to finding the largest
induced -colorable subgraph, which for is equivalent (by
complementation) to \textsc{Odd Cycle Transversal}.
We prove that for every fixed pattern graph without loops, \textsc{Max
Partial -Coloring} can be solved:
in -free graphs in polynomial time, whenever is a
threshold graph;
in -free graphs in polynomial time;
in -free graphs in time ;
in -free graphs in time
.
Here, is the number of vertices of the input graph and is
the maximum size of a clique in~. Furthermore, combining the mentioned
algorithms for -free and for -free
graphs with a simple branching procedure, we obtain subexponential-time
algorithms for \textsc{Max Partial -Coloring} in these classes of graphs.
Finally, we show that even a restricted variant of \textsc{Max Partial
-Coloring} is -hard in the considered subclasses of -free
graphs, if we allow loops on
- …