8,146 research outputs found

    Partial Complementation of Graphs

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    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

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    Let VV be a set of cardinality vv (possibly infinite). Two graphs GG and G′G' with vertex set VV are {\it isomorphic up to complementation} if G′G' is isomorphic to GG or to the complement Gˉ\bar G of GG. Let kk be a non-negative integer, GG and G′G' are {\it kk-hypomorphic up to complementation} if for every kk-element subset KK of VV, the induced subgraphs G_↾KG\_{\restriction K} and G′_↾KG'\_{\restriction K} are isomorphic up to complementation. A graph GG is {\it kk-reconstructible up to complementation} if every graph G′G' which is kk-hypomorphic to GG up to complementation is in fact isomorphic to GG up to complementation. We give a partial characterisation of the set S\mathcal S of pairs (n,k)(n,k) such that two graphs GG and G′G' on the same set of nn vertices are equal up to complementation whenever they are kk-hypomorphic up to complementation. We prove in particular that S\mathcal S contains all pairs (n,k)(n,k) such that 4≤k≤n−44\leq k\leq n-4. We also prove that 4 is the least integer kk such that every graph GG having a large number nn of vertices is kk-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

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    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

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    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

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    We define a new class of languages of ω\omega-words, strictly extending ω\omega-regular languages. One way to present this new class is by a type of regular expressions. The new expressions are an extension of ω\omega-regular expressions where two new variants of the Kleene star L∗L^* are added: LBL^B and LSL^S. 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 (aBb)ω(a^Bb)^\omega represents the language of infinite words over the letters a,ba,b where there is a common bound on the number of consecutive letters aa. The expression (aSb)ω(a^Sb)^\omega represents a similar language, but this time the distance between consecutive bb'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 LBL^B or LSL^S---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

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    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

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    We study the \textsc{Max Partial HH-Coloring} problem: given a graph GG, find the largest induced subgraph of GG that admits a homomorphism into HH, where HH is a fixed pattern graph without loops. Note that when HH is a complete graph on kk vertices, the problem reduces to finding the largest induced kk-colorable subgraph, which for k=2k=2 is equivalent (by complementation) to \textsc{Odd Cycle Transversal}. We prove that for every fixed pattern graph HH without loops, \textsc{Max Partial HH-Coloring} can be solved: ∙\bullet in {P5,F}\{P_5,F\}-free graphs in polynomial time, whenever FF is a threshold graph; ∙\bullet in {P5,bull}\{P_5,\textrm{bull}\}-free graphs in polynomial time; ∙\bullet in P5P_5-free graphs in time nO(ω(G))n^{\mathcal{O}(\omega(G))}; ∙\bullet in {P6,1-subdivided claw}\{P_6,\textrm{1-subdivided claw}\}-free graphs in time nO(ω(G)3)n^{\mathcal{O}(\omega(G)^3)}. Here, nn is the number of vertices of the input graph GG and ω(G)\omega(G) is the maximum size of a clique in~GG. Furthermore, combining the mentioned algorithms for P5P_5-free and for {P6,1-subdivided claw}\{P_6,\textrm{1-subdivided claw}\}-free graphs with a simple branching procedure, we obtain subexponential-time algorithms for \textsc{Max Partial HH-Coloring} in these classes of graphs. Finally, we show that even a restricted variant of \textsc{Max Partial HH-Coloring} is NP\mathsf{NP}-hard in the considered subclasses of P5P_5-free graphs, if we allow loops on HH
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