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Bibliographic Details
Main Author:	Gilbert, Jesse D.
Format:	Preprint
Published:	2011
Subjects:	Discrete Mathematics Combinatorics
Online Access:	https://arxiv.org/abs/1102.1087
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Using Graph Theory to Derive Inequalities for the Bell Numbers
by: Hertz, Alain, et al.
Published: (2021)

Fault tolerance for metric dimension and its variants
by: Geneson, Jesse, et al.
Published: (2025)

Computation of Graph Polynomials via Tree Decomposition: Theory, Algorithms, and Python Implementation
by: Bafna, Mehul, et al.
Published: (2025)

On Large Induced Outerplanar Subgraphs in $2$-Outerplanar Graphs
by: D'Elia, Marco, et al.
Published: (2026)

Structural Parameters for Dense Temporal Graphs
by: Enright, Jessica, et al.
Published: (2024)

Large Induced Subgraphs of Bounded Degree in Outerplanar and Planar Graphs
by: D'Elia, Marco, et al.
Published: (2024)

Stable non-minimal fixed points of threshold-linear networks
by: Geneson, Jesse
Published: (2025)

Theoretical and Computational Approaches to Determining Sets of Orders for $(k,g)$-Graphs
by: Eze, L. C., et al.
Published: (2025)

On a Problem of Ramsey Theory
by: Frasser, Carlos E.
Published: (2024)

Characterization of Circular-arc Graphs: III. Chordal Graphs
by: Cao, Yixin, et al.
Published: (2024)

A Characterization of Geodetic Graphs in Terms of their Embedded Even Graphs
by: Frasser, Carlos E.
Published: (2026)

Characterization of Chordal Circular-arc Graphs: I. Split Graphs
by: Cao, Yixin, et al.
Published: (2024)

Conflict-Free Coloring: Graphs of Bounded Clique Width and Intersection Graphs
by: Bhyravarapu, Sriram, et al.
Published: (2021)

On Threshold Compatibility Graphs
by: Hakim, Sheikh Azizul, et al.
Published: (2026)

Throttling for metric dimension and its variants
by: Brimkov, Boris, et al.
Published: (2025)

Line Graphs of Non-Word-Representable Graphs are Not Always Non-Word-Representable
by: Mozhui, Khyodeno, et al.
Published: (2025)

On Modular Edge Colourings of Graphs
by: Berthe, Gaétan, et al.
Published: (2025)

Word-Representation of Melon Graphs
by: Mozhui, Khyodeno, et al.
Published: (2026)

Enumerating Two-Orbit Graphs
by: Seka, David, et al.
Published: (2026)

Word-Representability of Shift Graphs
by: Roy, Suchanda, et al.
Published: (2026)

Bounds on Path Energy of Graphs
by: Narke, Amol P., et al.
Published: (2022)

Graph Reconstruction with Connectivity Queries
by: Kluk, Kacper, et al.
Published: (2024)

On Realizing Reconfiguration Graphs of Cliques
by: Hoang, Duc A.
Published: (2026)

On Generalizations of Pairwise Compatibility Graphs
by: Calamoneri, Tiziana, et al.
Published: (2021)

3-Colouring Planar Graphs
by: Dujmović, Vida, et al.
Published: (2025)

On a Characterization of Spartan Graphs
by: Misra, Neeldhara, et al.
Published: (2025)

On the Cop Number of String Graphs
by: Das, Sandip, et al.
Published: (2024)

Characterization of Split Comparability Graphs
by: Dwary, Tithi, et al.
Published: (2025)

On Tuza's Conjecture in Dense Graphs
by: Chahua, Luis, et al.
Published: (2024)

Vertex Ranking of Degenerate Graphs
by: Iacono, John, et al.
Published: (2024)

Ramanujan Graphs and Interlacing Families
by: Srivastava, Nikhil
Published: (2024)

Sparse String Graphs and Region Intersection Graphs over Minor-Closed Classes have Linear Expansion
by: Karol, Nikolai, et al.
Published: (2026)

Knowledge management in House of Graphs
by: Devillez, Gauvain, et al.
Published: (2026)

HIST-Critical Graphs and Malkevitch's Conjecture
by: Goedgebeur, Jan, et al.
Published: (2024)

Induced Minors and Region Intersection Graphs
by: Bonnet, Édouard, et al.
Published: (2025)

Counting Colored Tilings on Grids and Graphs
by: Ramírez, José L., et al.
Published: (2024)

Lower Bounds for Induced-Universal Graphs
by: Gavoille, Cyril, et al.
Published: (2025)

Geodetic Graphs: Experiments and New Constructions
by: Stober, Florian, et al.
Published: (2023)

On Separating Path and Tree Systems in Graphs
by: Biniaz, Ahmad, et al.
Published: (2023)

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// Anecdotal testing Today and Yesterday seems to indicate destroy() // is called by the garbage collector and everything is good. // Something to keep in mind though. return 0; } /** * Write function that is called when session data is to be saved. * * @param string $sessId The current session ID * @param string $data The session data to write * * @return bool */ public function write($sessId, $data): bool { if ($this->writesDisabled) { return true; } return $this->saveSession($sessId, $data); } /** * A function that is called internally when session data is to be saved. * * @param string $sessId The current session ID * @param string $data The session data to write * * @return bool */ abstract protected function saveSession($sessId, $data): bool; }

* * @return void */ public function writeClose() { // The assumption is that we're using PHP's ext/session. // session_write_close() will actually overwrite $_SESSION with an // empty array on completion -- which leads to a mismatch between what // is in the storage object and $_SESSION. To get around this, we // temporarily reset $_SESSION to an array, and then re-link it to // the storage object. // // Additionally, while you _can_ write to $_SESSION following a // session_write_close() operation, no changes made to it will be // flushed to the session handler. As such, we now mark the storage // object isImmutable. $storage = $this->getStorage(); if (! $storage->isImmutable()) { $_SESSION = $storage->toArray(true); session_write_close(); $storage->fromArray($_SESSION); $storage->markImmutable(); } } /** * Attempt to set the session name * * If the session has already been started, or if the name provided fails * validation, an exception will be raised. * * @param string $name * @return SessionManager * @throws Exception\InvalidArgumentException */ public function setName($name) { if ($this->sessionExists()) { throw new Exception\InvalidArgumentException( 'Cannot set session name after a session has already started'

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