Nonempty intersection of longest paths in graphs without forbidden pairs

In 1966, Gallai asked whether all longest paths in a connected graph share a common vertex. Counterexamples indicate that this is not true in general. However, Gallai's question is positive for certain well-known classes of connected graphs, such as split graphs, interval graphs, circular arc graphs, outerplanar graphs, and series-parallel graphs. A graph is $2K_2$-free if it does not contain two independent edges as an induced subgraph. In this short note, we show that, in nonempty $2K_2$-free graphs, every vertex of maximum degree is common to all longest paths. Our result implies that all longest paths in a nonempty $2K_2$-free graph have a nonempty intersection. In particular, it strengthens the result on split graphs, as split graphs are $2K_2$-free.

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A variational approach to the alternating projections method

Journal of Global Optimization ◽

10.1007/s10898-021-01025-y ◽

2021 ◽

Author(s):

Carlo Alberto De Bernardi ◽

Enrico Miglierina

Keyword(s):

Convex Sets ◽

Nonempty Intersection ◽

Feasibility Problem ◽

Alternating Projections ◽

Limit Sets ◽

Von Neumann ◽

Convex Feasibility ◽

Starting Point ◽

Method Of Alternating Projections ◽

Alternating Projections Method

AbstractThe 2-sets convex feasibility problem aims at finding a point in the nonempty intersection of two closed convex sets A and B in a Hilbert space H. The method of alternating projections is the simplest iterative procedure for finding a solution and it goes back to von Neumann. In the present paper, we study some stability properties for this method in the following sense: we consider two sequences of closed convex sets $$\{A_n\}$$ { A n } and $$\{B_n\}$$ { B n } , each of them converging, with respect to the Attouch-Wets variational convergence, respectively, to A and B. Given a starting point $$a_0$$ a 0 , we consider the sequences of points obtained by projecting on the “perturbed” sets, i.e., the sequences $$\{a_n\}$$ { a n } and $$\{b_n\}$$ { b n } given by $$b_n=P_{B_n}(a_{n-1})$$ b n = P B n ( a n - 1 ) and $$a_n=P_{A_n}(b_n)$$ a n = P A n ( b n ) . Under appropriate geometrical and topological assumptions on the intersection of the limit sets, we ensure that the sequences $$\{a_n\}$$ { a n } and $$\{b_n\}$$ { b n } converge in norm to a point in the intersection of A and B. In particular, we consider both when the intersection $$A\cap B$$ A ∩ B reduces to a singleton and when the interior of $$A \cap B$$ A ∩ B is nonempty. Finally we consider the case in which the limit sets A and B are subspaces.

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A new method for recognizing and locating objects by searching longest paths

[1992] Proceedings. 11th IAPR International Conference on Pattern Recognition ◽

10.1109/icpr.1992.201596 ◽

2003 ◽

Cited By ~ 1

Author(s):

F. Boussofiane ◽

G. Bertrand

Keyword(s):

New Method ◽

Longest Paths

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Making the hyperreal line both saturated and complete

Journal of Symbolic Logic ◽

10.2307/2275069 ◽

1991 ◽

Vol 56 (3) ◽

pp. 1016-1025 ◽

Cited By ~ 8

Author(s):

H. Jerome Keisler ◽

James H. Schmerl

Keyword(s):

Set Theory ◽

Nonempty Intersection ◽

Finite Intersection ◽

Second Order Arithmetic ◽

Ordered Field ◽

Finite Intersection Property ◽

Regular Cardinals ◽

Hyperreal Numbers ◽

Order Arithmetic ◽

Internal Sets

AbstractIn a nonstandard universe, the κ-saturation property states that any family of fewer than κ internal sets with the finite intersection property has a nonempty intersection. An ordered field F is said to have the λ-Bolzano-Weierstrass property iff F has cofinality λ and every bounded λ-sequence in F has a convergent λ-subsequence. We show that if κ < λ are uncountable regular cardinals and βα < λ whenever α < κ and β < λ then there is a κ-saturated nonstandard universe in which the hyperreal numbers have the λ-Bolzano-Weierstrass property. The result also applies to certain fragments of set theory and second order arithmetic.

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On Weak Contractive Cyclic Maps in Generalized Metric Spaces and Some Related Results on Best Proximity Points and Fixed Points

Discrete Dynamics in Nature and Society ◽

10.1155/2016/4186960 ◽

2016 ◽

Vol 2016 ◽

pp. 1-14 ◽

Cited By ~ 1

Author(s):

M. De la Sen

Keyword(s):

Fixed Points ◽

Metric Spaces ◽

Nonempty Intersection ◽

Generalized Metric Spaces ◽

Best Proximity Points ◽

Generalized Metric ◽

Cyclic Maps ◽

Convergence Of Sequences ◽

Composite Maps ◽

Decreasing Functions

This paper discusses the properties of convergence of sequences to limit cycles defined by best proximity points of adjacent subsets for two kinds of weak contractive cyclic maps defined by composite maps built with decreasing functions with either the so-calledr-weaker Meir-Keeler orr,r0-stronger Meir-Keeler functions in generalized metric spaces. Particular results about existence and uniqueness of fixed points are obtained for the case when the sets of the cyclic disposal have a nonempty intersection. Illustrative examples are discussed.

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Complexity of Nondeterministic Functions

BRICS Report Series ◽

10.7146/brics.v1i2.21668 ◽

1994 ◽

Vol 1 (2) ◽

Cited By ~ 3

Author(s):

Alexander E. Andreev

Keyword(s):

Upper Bound ◽

Randomized Algorithms ◽

Nonempty Intersection ◽

Linear Size ◽

Worst Case ◽

K Value ◽

Hitting Set ◽

Set Size

The complexity of a nondeterministic function is the minimum possible complexity of its determinisation. The entropy of a nondeterministic function, F, is minus the logarithm of the ratio between the number of determinisations of F and the number of all deterministic functions. We obtain an upper bound on the complexity of a nondeterministic function with restricted entropy for the worst case. These bounds have strong applications in the problem of algorithm derandomization. A lot of randomized algorithms can be converted to deterministic ones if we have an effective hitting set with certain parameters (a set is hitting for a set system if it has a nonempty intersection with any set from the system). Linial, Luby, Saks and Zuckerman (1993) constructed the best effective hitting set for the system of k-value, n-dimensional rectangles. The set size is polynomial in k log n / epsilon. Our bounds of nondeterministic functions complexity offer a possibility to construct an effective hitting set for this system with almost linear size in k log n / epsilon.

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