On the Largest Dynamic Monopolies of Graphs with a Given Average Threshold

AbstractLet G be a graph and let τ be an assignment of nonnegative integer thresholds to the vertices of G. A subset of vertices, D, is said to be a τ-dynamicmonopoly if V(G) can be partitioned into subsets D0 , D1, …, Dk such that D0 = D and for any i ∊ {0, . . . , k−1}, each vertex v in Di+1 has at least τ(v) neighbors in D0∪··· ∪Di. Denote the size of smallest τ-dynamicmonopoly by dynτ(G) and the average of thresholds in τ by τ. We show that the values of dynτ(G) over all assignments τ with the same average threshold is a continuous set of integers. For any positive number t, denote the maximum dynτ(G) taken over all threshold assignments τ with τ ≤ t, by Ldynt(G). In fact, Ldynt(G) shows the worst-case value of a dynamicmonopoly when the average threshold is a given number t. We investigate under what conditions on t, there exists an upper bound for Ldynt(G) of the form c|G|, where c < 1. Next, we show that Ldynt(G) is coNP-hard for planar graphs but has polynomial-time solution for forests.

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On $t$-Common List-Colorings

The Electronic Journal of Combinatorics ◽

10.37236/6738 ◽

2017 ◽

Vol 24 (3) ◽

Author(s):

Hojin Choi ◽

Young Soo Kwon

Keyword(s):

Positive Integer ◽

Planar Graph ◽

Chromatic Number ◽

Upper Bound ◽

Nonnegative Integer ◽

Planar Graphs ◽

Four Color Theorem ◽

List Colorings ◽

List Chromatic Number ◽

Positive Integers

In this paper, we introduce a new variation of list-colorings. For a graph $G$ and for a given nonnegative integer $t$, a $t$-common list assignment of $G$ is a mapping $L$ which assigns each vertex $v$ a set $L(v)$ of colors such that given set of $t$ colors belong to $L(v)$ for every $v\in V(G)$. The $t$-common list chromatic number of $G$ denoted by $ch_t(G)$ is defined as the minimum positive integer $k$ such that there exists an $L$-coloring of $G$ for every $t$-common list assignment $L$ of $G$, satisfying $|L(v)| \ge k$ for every vertex $v\in V(G)$. We show that for all positive integers $k, \ell$ with $2 \le k \le \ell$ and for any positive integers $i_1 , i_2, \ldots, i_{k-2}$ with $k \le i_{k-2} \le \cdots \le i_1 \le \ell$, there exists a graph $G$ such that $\chi(G)= k$, $ch(G) = \ell$ and $ch_t(G) = i_t$ for every $t=1, \ldots, k-2$. Moreover, we consider the $t$-common list chromatic number of planar graphs. From the four color theorem and the result of Thomassen (1994), for any $t=1$ or $2$, the sharp upper bound of $t$-common list chromatic number of planar graphs is $4$ or $5$. Our first step on $t$-common list chromatic number of planar graphs is to find such a sharp upper bound. By constructing a planar graph $G$ such that $ch_1(G) =5$, we show that the sharp upper bound for $1$-common list chromatic number of planar graphs is $5$. The sharp upper bound of $2$-common list chromatic number of planar graphs is still open. We also suggest several questions related to $t$-common list chromatic number of planar graphs.

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An improved upper bound on the linear 2-arboricity of planar graphs

Discrete Mathematics ◽

10.1016/j.disc.2015.07.003 ◽

2016 ◽

Vol 339 (1) ◽

pp. 39-45 ◽

Cited By ~ 4

Author(s):

Yiqiao Wang

Keyword(s):

Upper Bound ◽

Planar Graphs

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REDUCTION TREE OF THE BINARY GENERALIZED POST CORRESPONDENCE PROBLEM

International Journal of Foundations of Computer Science ◽

10.1142/s0129054111008143 ◽

2011 ◽

Vol 22 (02) ◽

pp. 473-490 ◽

Cited By ~ 4

Author(s):

VESA HALAVA ◽

ŠTĚPÁN HOLUB

Keyword(s):

Detailed Analysis ◽

Polynomial Time ◽

Upper Bound ◽

Decision Process ◽

Correspondence Problem ◽

Single Instance ◽

Binary Case

An instance of the (Generalized) Post Correspondence Problem is during the decision process typically reduced to one or more other instances, called its successors. In this paper we study the reduction tree of GPCP in the binary case. This entails in particular a detailed analysis of the structure of end blocks. We give an upper bound for the number of end blocks, and show that even if an instance has more than one successor, it can nevertheless be reduced to a single instance. This, in particular, implies that binary GPCP can be decided in polynomial time.

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An $$O(n^{\epsilon })$$ Space and Polynomial Time Algorithm for Reachability in Directed Layered Planar Graphs

Algorithms and Computation - Lecture Notes in Computer Science ◽

10.1007/978-3-662-48971-0_52 ◽

2015 ◽

pp. 614-624 ◽

Cited By ~ 1

Author(s):

Diptarka Chakraborty ◽

Raghunath Tewari

Keyword(s):

Polynomial Time ◽

Planar Graphs ◽

Polynomial Time Algorithm ◽

Time Algorithm

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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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Stochastic Flips on Dimer Tilings

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2803 ◽

2010 ◽

Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽

Author(s):

Thomas Fernique ◽

Damien Regnault

Keyword(s):

Fixed Point ◽

Markov Process ◽

Upper Bound ◽

Numerical Experiments ◽

Triangular Grid ◽

Expected Number ◽

Worst Case ◽

Average Case ◽

International Audience

International audience This paper introduces a Markov process inspired by the problem of quasicrystal growth. It acts over dimer tilings of the triangular grid by randomly performing local transformations, called $\textit{flips}$, which do not increase the number of identical adjacent tiles (this number can be thought as the tiling energy). Fixed-points of such a process play the role of quasicrystals. We are here interested in the worst-case expected number of flips to converge towards a fixed-point. Numerical experiments suggest a $\Theta (n^2)$ bound, where $n$ is the number of tiles of the tiling. We prove a $O(n^{2.5})$ upper bound and discuss the gap between this bound and the previous one. We also briefly discuss the average-case.

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