scholarly journals The Separator Theorem for Rooted Directed Vertex Graphs

2001 ◽  
Vol 81 (1) ◽  
pp. 156-162 ◽  
Author(s):  
B.S. Panda
Keyword(s):  
Separator Theorem

Applications of a Planar Separator Theorem

1980 ◽  
Vol 9 (3) ◽  
pp. 615-627 ◽  
Author(s):  
Richard J. Lipton ◽  
Robert Endre Tarjan
Keyword(s):  
Separator Theorem

scholarly journals A Separator Theorem for Hypergraphs and a CSP-SAT Algorithm

2021 ◽  
Vol Volume 17, Issue 4 ◽  
Author(s):  
Michal Koucký ◽  
Vojtěch Rödl ◽  
Navid Talebanfard
Keyword(s):  
Deterministic Algorithm ◽  
Vertex Degree ◽  
Connected Components ◽  
Variable Frequency ◽  
Uniform Hypergraph ◽  
Separator Theorem

We show that for every $r \ge 2$ there exists $\epsilon_r > 0$ such that any $r$-uniform hypergraph with $m$ edges and maximum vertex degree $o(\sqrt{m})$ contains a set of at most $(\frac{1}{2} - \epsilon_r)m$ edges the removal of which breaks the hypergraph into connected components with at most $m/2$ edges. We use this to give an algorithm running in time $d^{(1 - \epsilon_r)m}$ that decides satisfiability of $m$-variable $(d, k)$-CSPs in which every variable appears in at most $r$ constraints, where $\epsilon_r$ depends only on $r$ and $k\in o(\sqrt{m})$. Furthermore our algorithm solves the corresponding #CSP-SAT and Max-CSP-SAT of these CSPs. We also show that CNF representations of unsatisfiable $(2, k)$-CSPs with variable frequency $r$ can be refuted in tree-like resolution in size $2^{(1 - \epsilon_r)m}$. Furthermore for Tseitin formulas on graphs with degree at most $k$ (which are $(2, k)$-CSPs) we give a deterministic algorithm finding such a refutation.

scholarly journals A Separator Theorem for String Graphs and its Applications

2009 ◽  
Vol 19 (3) ◽  
pp. 371-390 ◽  
Author(s):  
JACOB FOX ◽  
JÁNOS PACH
Keyword(s):  
Intersection Graph ◽  
Equal Size ◽  
Bipartite Subgraph ◽  
String Graphs ◽  
String Graph ◽  
Complete Bipartite Subgraph ◽  
Labelled Trees ◽  
Separator Theorem ◽  
Labelled Graphs

A string graph is the intersection graph of a collection of continuous arcs in the plane. We show that any string graph with m edges can be separated into two parts of roughly equal size by the removal of $O(m^{3/4}\sqrt{\log m})$ vertices. This result is then used to deduce that every string graph with n vertices and no complete bipartite subgraph Kt,t has at most ctn edges, where ct is a constant depending only on t. Another application shows that locally tree-like string graphs are globally tree-like: for any ε > 0, there is an integer g(ε) such that every string graph with n vertices and girth at least g(ε) has at most (1 + ε)n edges. Furthermore, the number of such labelled graphs is at most (1 + ε)nT(n), where T(n) = nn−2 is the number of labelled trees on n vertices.

A Separator Theorem for Planar Graphs

1979 ◽  
Vol 36 (2) ◽  
pp. 177-189 ◽  
Author(s):  
Richard J. Lipton ◽  
Robert Endre Tarjan
Keyword(s):  
Planar Graphs ◽  
Separator Theorem

scholarly journals A Separator Theorem for Chordal Graphs

1984 ◽  
Vol 5 (3) ◽  
pp. 306-313 ◽  
Author(s):  
John R. Gilbert ◽  
Donald J. Rose ◽  
Anders Edenbrandt
Keyword(s):  
Chordal Graphs ◽  
Separator Theorem

scholarly journals Applications of a New Separator Theorem for String Graphs

2013 ◽  
Vol 23 (1) ◽  
pp. 66-74 ◽  
Author(s):  
JACOB FOX ◽  
JÁNOS PACH
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Present Note ◽  
Intersection Graph ◽  
Type Result ◽  
Vertex Separator ◽  
String Graphs ◽  
String Graph ◽  
Ramsey Type ◽  
Separator Theorem

An intersection graph of curves in the plane is called astring graph. Matoušek almost completely settled a conjecture of the authors by showing that every string graph withmedges admits a vertex separator of size$O(\sqrt{m}\log m)$. In the present note, this bound is combined with a result of the authors, according to which every dense string graph contains a large complete balanced bipartite graph. Three applications are given concerning string graphsGwithnvertices: (i) ifKt⊈Gfor somet, then the chromatic number ofGis at most (logn)O(logt); (ii) ifKt,t⊈G, thenGhas at mostt(logt)O(1)nedges,; and (iii) a lopsided Ramsey-type result, which shows that the Erdős–Hajnal conjecture almost holds for string graphs.

Applications of a planar separator theorem

1977 ◽  
Author(s):  
Richard J. Lipton ◽  
Robert Endre Tarjan
Keyword(s):  
Separator Theorem
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