اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدReconfiguration of Connected Graph Partitions
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Motivated by recent computational models for redistricting and detection of gerrymandering, we study the following problem on graph partitions. Given a graph $G$ and an integer $kgeq 1$, a $k$-district map of $G$ is a partition of $V(G)$ into $k$ nonempty subsets, called districts, each of which induces a connected subgraph of $G$. A switch is an operation that modifies a $k$-district map by reassigning a subset of vertices from one district to an adjacent district; a 1-switch is a switch that moves a single vertex. We study the connectivity of the configuration space of all $k$-district maps of a graph $G$ under 1-switch operations. We give a combinatorial characterization for the connectedness of this space that can be tested efficiently. We prove that it is NP-complete to decide whether there exists a sequence of 1-switches that takes a given $k$-district map into another; and NP-hard to find the shortest such sequence (even if a sequence of polynomial length is known to exist). We also present efficient algorithms for computing a sequence of 1-switches that takes a given $k$-district map into another when the space is connected, and show that these algorithms perform a worst-case optimal number of switches up to constant factors.
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Motivated by applications in gerrymandering detection, we study a reconfiguration problem on connected partitions of a connected graph $G$. A partition of $V(G)$ is emph{connected} if every part induces a connected subgraph. In many applications, it
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