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Isolde Adler

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7 papers
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7

MFCS Conference 2024 Conference Paper

Monotonicity of the Cops and Robber Game for Bounded Depth Treewidth

  • Isolde Adler
  • Eva Fluck

We study a variation of the cops and robber game characterising treewidth, where in each round at most one cop may be placed and in each play at most q rounds are played, where q is a parameter of the game. We prove that if k cops have a winning strategy in this game, then k cops have a monotone winning strategy. As a corollary we obtain a new characterisation of bounded depth treewidth, and we give a positive answer to an open question by Fluck, Seppelt and Spitzer (2024), thus showing that graph classes of bounded depth treewidth are homomorphism distinguishing closed. Our proof of monotonicity substantially reorganises a winning strategy by first transforming it into a pre-tree decomposition, which is inspired by decompositions of matroids, and then applying an intricate breadth-first "cleaning up" procedure along the pre-tree decomposition (which may temporarily lose the property of representing a strategy), in order to achieve monotonicity while controlling the number of rounds simultaneously across all branches of the decomposition via a vertex exchange argument. We believe this can be useful in future research.

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SODA Conference 2021 Conference Paper

On Testability of First-Order Properties in Bounded-Degree Graphs

  • Isolde Adler
  • Noleen Köhler
  • Pan Peng 0001

We study property testing of properties that are definable in first-order logic (FO) in the bounded-degree graph and relational structure models. We show that any FO property that is defined by a formula with quantifier prefix ∃∗∀∗ is testable (i. e. , testable with constant query complexity), while there exists an FO property that is expressible by a formula with quantifier prefix ∀∗∃∗ that is not testable. In the dense graph model, a similar picture is long known (Alon, Fischer, Krivelevich, Szegedy, Combinatorica 2000), despite the very different nature of the two models. In particular, we obtain our lower bound by a first-order formula that defines a class of bounded-degree expanders, based on zig-zag products of graphs. We expect this to be of independent interest. We then prove testability of some first-order properties that speak about isomorphism types of neighbourhoods, including testability of 1-neighbourhood-freeness, and r -neighbourhood-freeness under a mild assumption on the degrees.

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CSL Conference 2009 Conference Paper

Tree-Width for First Order Formulae

  • Isolde Adler
  • Mark Weyer

Abstract We introduce tree-width for first order formulae ϕ, fotw( ϕ ). We show that computing fotw is fixed-parameter tractable with parameter fotw. Moreover, we show that on classes of formulae of bounded fotw, model checking is fixed parameter tractable, with parameter the length of the formula. This is done by translating a formula ϕ with fotw( ϕ ) < k into a formula of the k -variable fragment \({\mathcal L^k}\) of first order logic. For fixed k, the question whether a given first order formula is equivalent to an \({\mathcal L^k}\) formula is undecidable. In contrast, the classes of first order formulae with bounded fotw are fragments of first order logic for which the equivalence is decidable. Our notion of tree-width generalises tree-width of conjunctive queries to arbitrary formulae of first order logic by taking into account the quantifier interaction in a formula. Moreover, it is more powerful than the notion of elimination-width of quantified constraint formulae, defined by Chen and Dalmau (CSL 2005): For quantified constraint formulae, both bounded elimination-width and bounded fotw allow for model checking in polynomial time. We prove that fotw of a quantified constraint formula φ is bounded by the elimination-width of φ, and we exhibit a class of quantified constraint formulae with bounded fotw, that has unbounded elimination-width. A similar comparison holds for strict tree-width of non-recursive stratified datalog as defined by Flum, Frick, and Grohe (JACM 49, 2002). Finally, we show that fotw has a characterization in terms of a robber and cops game without monotonicity cost.

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