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DTSTART;TZID=Asia/Seoul:20210317T170000
DTEND;TZID=Asia/Seoul:20210317T180000
DTSTAMP:20260420T175531
CREATED:20210228T115822Z
LAST-MODIFIED:20240705T190042Z
UID:3692-1616000400-1616004000@dimag.ibs.re.kr
SUMMARY:Yixin Cao (操宜新)\, Recognizing (unit) interval graphs by zigzag graph searches
DESCRIPTION:Corneil\, Olariu\, and Stewart [SODA 1998; SIAM Journal on Discrete Mathematics 2009] presented a recognition algorithm for interval graphs by six graph searches. Li and Wu [Discrete Mathematics & Theoretical Computer Science 2014] simplified it to only four. The great simplicity of the latter algorithm is however eclipsed by the complicated and long proofs. The main purpose of this paper is to present a new and significantly shorter proof for Li and Wu’s algorithm\, as well as a simpler implementation. We also give a self-contained presentation of the recognition algorithm of Corneil [Discrete Applied Mathematics 2004] for unit interval graphs\, based on three sweeps of graph searches. Moreover\, we show that two sweeps are already sufficient. Toward the proofs of the main results\, we make several new structural observations that might be of independent interests.
URL:https://dimag.ibs.re.kr/event/2021-03-17/
LOCATION:Room B232\, IBS (기초과학연구원)
CATEGORIES:Virtual Discrete Math Colloquium
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DTSTART;TZID=Asia/Seoul:20210324T170000
DTEND;TZID=Asia/Seoul:20210324T180000
DTSTAMP:20260420T175531
CREATED:20210219T024236Z
LAST-MODIFIED:20240705T191012Z
UID:3649-1616605200-1616608800@dimag.ibs.re.kr
SUMMARY:Édouard Bonnet\, Twin-width and ordered binary structures
DESCRIPTION:The twin-width of a graph G can be defined as the least integer d such that there is a sequence of length |V(G)| of (strictly) coarser and coarser partitions of its vertex set V(G)\, and every part X of every partition P of the sequence has at most d other parts Y of P with both at least one edge and at least one non-edge between X and Y.  Twin-width is closely tied to total orders on the vertices\, and can be extended to general binary structures. We will thus consider the twin-width of ordered binary structures\, or if you prefer\, matrices on a finite alphabet. This turns out to be key in understanding combinatorial\, algorithmic\, and model-theoretic properties of (hereditary) classes of those objects. We will see several characterizations of bounded twin-width for these classes. The main consequences in the three domains read as follows. \n\nEnumerative combinatorics: All the classes of 0\,1-matrices with superexponential growth have growth at least n! (in turn resolving a conjecture of Balogh\, Bollobás\, and Morris on the growth of hereditary classes of ordered graphs).\nAlgorithms: First-order model checking of ordered binary structures is tractable exactly when the twin-width is bounded.\nFinite model theory: Monadically-dependent and dependent hereditary classes of ordered binary structures are the same.\n\nIn addition we get a fixed-parameter algorithm approximating matrix twin-width within a function of the optimum\, which is still missing for unordered graphs. \nJoint work with Ugo Giocanti\, Patrice Ossona de Mendez\, and Stéphan Thomassé. Similar results have been obtained independently by Pierre Simon and Szymon Toruńczyk.
URL:https://dimag.ibs.re.kr/event/2021-03-24/
LOCATION:Zoom ID: 869 4632 6610 (ibsdimag)
CATEGORIES:Virtual Discrete Math Colloquium
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