scholarly journals Balanced Avoidance Games on Random Graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Martin Marciniszyn ◽  
Dieter Mitsche ◽  
Miloš Stojaković
Keyword(s):  
Random Graphs ◽  
Complete Graph ◽  
Optimal Strategy ◽  
Random Order ◽  
The Other ◽  
Threshold Function ◽  
Graph Theoretic ◽  
International Audience ◽  
Balanced Coloring ◽  
Monochromatic Copy

International audience We introduce and study balanced online graph avoidance games on the random graph process. The game is played by a player we call Painter. Edges of the complete graph with $n$ vertices are revealed two at a time in a random order. In each move, Painter immediately and irrevocably decides on a balanced coloring of the new edge pair: either the first edge is colored red and the second one blue or vice versa. His goal is to avoid a monochromatic copy of a given fixed graph $H$ in both colors for as long as possible. The game ends as soon as the first monochromatic copy of $H$ has appeared. We show that the duration of the game is determined by a threshold function $m_H = m_H(n)$. More precisely, Painter will asymptotically almost surely (a.a.s.) lose the game after $m = \omega (m_H)$ edge pairs in the process. On the other hand, there is an essentially optimal strategy, that is, if the game lasts for $m = o(m_H)$ moves, then Painter will a.a.s. successfully avoid monochromatic copies of H using this strategy. Our attempt is to determine the threshold function for certain graph-theoretic structures, e.g., cycles.

scholarly journals Parameterized Problems Related to Seidel's Switching

2011 ◽  
Vol Vol. 13 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Eva Jelinkova ◽  
Ondrej Suchy ◽  
Petr Hlineny ◽  
Jan Kratochvil
Keyword(s):  
Minimum Degree ◽  
The Other ◽  
Complete Graphs ◽  
Maximum Likelihood Decoding ◽  
Induced Subgraph ◽  
Fixed Parameter Tractable ◽  
Graph Theoretic ◽  
Np Completeness ◽  
Fixed Parameter ◽  
International Audience

Graphs and Algorithms International audience Seidel's switching is a graph operation which makes a given vertex adjacent to precisely those vertices to which it was non-adjacent before, while keeping the rest of the graph unchanged. Two graphs are called switching-equivalent if one can be made isomorphic to the other by a sequence of switches. In this paper, we continue the study of computational complexity aspects of Seidel's switching, concentrating on Fixed Parameter Complexity. Among other results we show that switching to a graph with at most k edges, to a graph of maximum degree at most k, to a k-regular graph, or to a graph with minimum degree at least k are fixed parameter tractable problems, where k is the parameter. On the other hand, switching to a graph that contains a given fixed graph as an induced subgraph is W [1]-complete. We also show the NP-completeness of switching to a graph with a clique of linear size, and of switching to a graph with small number of edges. A consequence of the latter result is the NP-completeness of Maximum Likelihood Decoding of graph theoretic codes based on complete graphs.

scholarly journals Manipulative Waiters with Probabilistic Intuition

2015 ◽  
Vol 25 (6) ◽  
pp. 823-849 ◽  
Author(s):  
MAŁGORZATA BEDNARSKA-BZDȨGA ◽  
DAN HEFETZ ◽  
MICHAEL KRIVELEVICH ◽  
TOMASZ ŁUCZAK
Keyword(s):  
Phase Transition ◽  
Random Graphs ◽  
Complete Graph ◽  
Absolute Constant ◽  
Perfect Information ◽  
Graph Theoretic ◽  
Graph Property ◽  
Positive Integers ◽  
Monotone Graph ◽  
Transition Type

For positive integersnandqand a monotone graph property$\mathcal{A}$, we consider the two-player, perfect information game WC(n,q,$\mathcal{A}$), which is defined as follows. The game proceeds in rounds. In each round, the first player, called Waiter, offers the second player, called Client,q+ 1 edges of the complete graphKnwhich have not been offered previously. Client then chooses one of these edges which he keeps and the remainingqedges go back to Waiter. If, at the end of the game, the graph which consists of the edges chosen by Client satisfies the property$\mathcal{A}$, then Waiter is declared the winner; otherwise Client wins the game. In this paper we study such games (also known as Picker–Chooser games) for a variety of natural graph-theoretic parameters, such as the size of a largest component or the length of a longest cycle. In particular, we describe a phase transition type phenomenon which occurs when the parameterqis close tonand is reminiscent of phase transition phenomena in random graphs. Namely, we prove that ifq⩾ (1 + ϵ)n, then Client can avoid components of ordercϵ−2lnnfor some absolute constantc> 0, whereas forq⩽ (1 − ϵ)n, Waiter can force a giant, linearly sized component in Client's graph. In the second part of the paper, we prove that Waiter can force Client's graph to be pancyclic for everyq⩽cn, wherec> 0 is an appropriate constant. Note that this behaviour is in stark contrast to the threshold for pancyclicity and Hamiltonicity of random graphs.

scholarly journals Toppling numbers of complete and random graphs

2014 ◽  
Vol Vol. 16 no. 3 (Graph Theory) ◽  
Author(s):  
Anthony Bonato ◽  
William B. Kinnersley ◽  
Pawel Pralat
Keyword(s):  
Graph Theory ◽  
Differential Equations ◽  
Random Graphs ◽  
Complete Graph ◽  
Complete Graphs ◽  
Asymptotic Bounds ◽  
Large N ◽  
Chip Firing ◽  
International Audience ◽  
Person Game

Graph Theory International audience We study a two-person game played on graphs based on the widely studied chip-firing game. Players Max and Min alternately place chips on the vertices of a graph. When a vertex accumulates as many chips as its degree, it fires, sending one chip to each neighbour; this may in turn cause other vertices to fire. The game ends when vertices continue firing forever. Min seeks to minimize the number of chips played during the game, while Max seeks to maximize it. When both players play optimally, the length of the game is the toppling number of a graph G, and is denoted by t(G). By considering strategies for both players and investigating the evolution of the game with differential equations, we provide asymptotic bounds on the toppling number of the complete graph. In particular, we prove that for sufficiently large n 0.596400 n2 < t(Kn) < 0.637152 n2. Using a fractional version of the game, we couple the toppling numbers of complete graphs and the binomial random graph G(n,p). It is shown that for pn ≥n² / √ log(n) asymptotically almost surely t(G(n,p))=(1+o(1)) p t(Kn).

scholarly journals Popular Matchings in Complete Graphs

Algorithmica ◽  
2021 ◽  
Author(s):  
Ágnes Cseh ◽  
Telikepalli Kavitha
Keyword(s):  
Complete Graph ◽  
The Other ◽  
Complete Graphs ◽  
Matching Problem ◽  
Graph Theoretic ◽  
The Given ◽  
Popular Matching

AbstractOur input is a complete graph G on n vertices where each vertex has a strict ranking of all other vertices in G. The goal is to construct a matching in G that is popular. A matching M is popular if M does not lose a head-to-head election against any matching $$M'$$ M ′ : here each vertex casts a vote for the matching in $$\{M,M'\}$$ { M , M ′ } in which it gets a better assignment. Popular matchings need not exist in the given instance G and the popular matching problem is to decide whether one exists or not. The popular matching problem in G is easy to solve for odd n. Surprisingly, the problem becomes $$\texttt {NP}$$ NP -complete for even n, as we show here. This is one of the few graph theoretic problems efficiently solvable when n has one parity and $$\texttt {NP}$$ NP -complete when n has the other parity.

scholarly journals The Ramsey number of books

2019 ◽  
Author(s):  
David Conlon
Keyword(s):  
Complete Graph ◽  
Edge Coloring ◽  
The Other ◽  
Ramsey Number ◽  
Natural Question ◽  
Probabilistic Argument ◽  
Complete Subgraph ◽  
Random Construction ◽  
Lower Order Terms ◽  
Monochromatic Copy

Ramsey's Theorem is among the most well-known results in combinatorics. The theorem states that any two-edge-coloring of a sufficiently large complete graph contains a large monochromatic complete subgraph. Indeed, any two-edge-coloring of a complete graph with N=4t−o(t) vertices contains a monochromatic copy of Kt. On the other hand, a probabilistic argument yields that there exists a two-edge-coloring of a complete graph with N=2t/2+o(t) with no monochromatic copy of Kt. Much attention has been paid to improving these classical bounds but only improvements to lower order terms have been obtained so far. A natural question in this setting is to ask whether every two-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of Kt that can be extended in many ways to a monochromatic copy of Kt+1. Specifically, Erdős, Faudree, Rousseau and Schelp in the 1970's asked whether every two-edge-coloring of KN contains a monochromatic copy of Kt that can be extended in at least (1−ok(1))2−tN ways to a monochromatic copy of Kt+1. A random two-edge-coloring of KN witnesses that this would be best possible. While the intuition coming from random constructions can be misleading, for example, a famous construction by Thomason shows the existence of a two-edge-coloring of a complete graph with fewer monochromatic copies of Kt than a random two-edge-coloring, this paper confirms that the intuition coming from a random construction is correct in this case. In particular, the author answers this question of Erdős et al. in the affirmative. The question can be phrased in the language of Ramsey theory as a problem on determining the Ramsey number of book graphs. A book graph B(k)t is a graph obtained from Kt by adding k new vertices and joining each new vertex to all the vertices of Kt. The main result of the paper asserts that every two-edge-coloring of a complete graph with N=2kt+ok(t) vertices contains a monochromatic copy of B(k)t.

scholarly journals Ramsey-type problems in orientations of graphs ⇤

2018 ◽  
Author(s):  
Bruno Pasqualotto Cavalar
Keyword(s):  
Random Graphs ◽  
Oriented Graph ◽  
Ramsey Number ◽  
Threshold Function ◽  
Ramsey Problem ◽  
Minimum Number ◽  
Ramsey Type ◽  
Monochromatic Copy

The Ramsey number R(H) of a graph H is the minimum number n such that there exists a graph G on n vertices with the property that every two-coloring of its edges contains a monochromatic copy of H. In this work we study a variant of this notion, called the oriented Ramsey problem, for an acyclic oriented graph H~ , in which we require that every orientation G~ of the graph G contains a copy of H~ . We also study the threshold function for this problem in random graphs. Finally, we consider the isometric case, in which we require the copy to be isometric, by which we mean that, for every two vertices x, y 2 V (H~ ) and their respective copies x0, y0 in G~ , the distance between x and y is equal to the distance between x0 and y0.

Expressive Microstructure in Music: A Preliminary Perceptual Assessment of Four Composers' "Pulses"

1989 ◽  
Vol 6 (3) ◽  
pp. 243-273 ◽  
Author(s):  
Bruno H. Repp
Keyword(s):  
Classical Music ◽  
Random Order ◽  
Preliminary Test ◽  
The Other ◽  
Musical Experience ◽  
General Music ◽  
Cyclic Patterns ◽  
Computer Controlled ◽  
Perceptual Assessment ◽  
Essential Prerequisite

According to a provocative theory set forth by Manfred Clynes, there are composer-specific cyclic patterns of (unnotated) musical microstructure that, when discovered and realized by a performer, help to give the music its characteristic expressive quality. Clynes, relying mainly on his own judgment as an experienced musician, has derived such personal "pulses" for several famous composers by imposing time and amplitude perturbations on computer-controlled performances of classical music and modifying them until they converged on some optimal expression. To conduct a preliminary test of the general music lover's appreciation of such "pulsed" performances, two sets of piano pieces by Beethoven, Haydn, Mozart, and Schubert, one in quadruple and the other in triple meter, were selected for this study. Each piece was synthesized with each composer's pulse and also without any pulse. These different versions were presented in random order to listeners of varying musical sophistication for preference judgments relative to the unpulsed version. There were reliable changes in listeners' pulse preferences across different composers' pieces, which affirms one essential prerequisite of Clynes' theory. Moreover, in several instances the "correct" pulse was preferred most, which suggests not only that these pulse patterns indeed capture composer- specific qualities, but also that listeners without extensive musical experience can appreciate them. In other cases, however, listeners' preferences were not as expected, and possible causes for these deviations are discussed.

scholarly journals Integrative statistical analyses of multiple liquid biopsy analytes in metastatic breast cancer

2021 ◽  
Vol 13 (1) ◽  
Author(s):  
Corinna Keup ◽  
Vinay Suryaprakash ◽  
Siegfried Hauch ◽  
Markus Storbeck ◽  
Peter Hahn ◽  
...  
Keyword(s):  
Breast Cancer ◽  
Overall Survival ◽  
Metastatic Breast Cancer ◽  
Mutual Information ◽  
Hierarchical Clustering ◽  
Liquid Biopsy ◽  
Metastatic Breast ◽  
The Other ◽  
Theoretic Analysis ◽  
Graph Theoretic

Abstract Background Single liquid biopsy analytes (LBAs) have been utilized for therapy selection in metastatic breast cancer (MBC). We performed integrative statistical analyses to examine the clinical relevance of using multiple LBAs: matched circulating tumor cell (CTC) mRNA, CTC genomic DNA (gDNA), extracellular vesicle (EV) mRNA, and cell-free DNA (cfDNA). Methods Blood was drawn from 26 hormone receptor-positive, HER2-negative MBC patients. CTC mRNA and EV mRNA were analyzed using a multi-marker qPCR. Plasma from CTC-depleted blood was utilized for cfDNA isolation. gDNA from CTCs was isolated from mRNA-depleted CTC lysates. CTC gDNA and cfDNA were analyzed by targeted sequencing. Hierarchical clustering was performed within each analyte, and its results were combined into a score termed Evaluation of multiple Liquid biopsy analytes In Metastatic breast cancer patients All from one blood sample (ELIMA.score), which calculates the contribution of each analyte to the overall survival prediction. Singular value decomposition (SVD), mutual information calculation, k-means clustering, and graph-theoretic analysis were conducted to elucidate the dependence between individual analytes. Results A combination of two/three/four LBAs increased the prevalence of patients with actionable signals. Aggregating the results of hierarchical clustering of individual LBAs into the ELIMA.score resulted in a highly significant correlation with overall survival, thereby bolstering evidence for the additive value of using multiple LBAs. Computation of mutual information indicated that none of the LBAs is independent of the others, but the ability of a single LBA to describe the others is rather limited—only CTC gDNA could partially describe the other three LBAs. SVD revealed that the strongest singular vectors originate from all four LBAs, but a majority originated from CTC gDNA. After k-means clustering of patients based on parameters of all four LBAs, the graph-theoretic analysis revealed CTC ERBB2 variants only in patients belonging to one particular cluster. Conclusions The additional benefits of using all four LBAs were objectively demonstrated in this pilot study, which also indicated a relative dominance of CTC gDNA over the other LBAs. Consequently, a multi-parametric liquid biopsy approach deconvolutes the genomic and transcriptomic complexity and should be considered in clinical practice.

scholarly journals Online Ramsey Games in Random Graphs

2009 ◽  
Vol 18 (1-2) ◽  
pp. 271-300 ◽  
Author(s):  
MARTIN MARCINISZYN ◽  
RETO SPÖHEL ◽  
ANGELIKA STEGER
Keyword(s):  
Lower Bound ◽  
Random Graphs ◽  
Empty Graph ◽  
Current Graph ◽  
Player Game ◽  
Monochromatic Copy

Consider the following one-player game. Starting with the empty graph onnvertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one ofravailable colours. The player's goal is to avoid creating a monochromatic copy of some fixed graphFfor as long as possible. We prove a lower bound ofnβ(F,r)on the typical duration of this game, where β(F,r) is a function that is strictly increasing inrand satisfies limr→∞β(F,r) = 2 − 1/m2(F), wheren2−1/m2(F)is the threshold of the corresponding offline colouring problem.

Oxidation of combined ingestion of glucose and fructose during exercise

2004 ◽  
Vol 96 (4) ◽  
pp. 1277-1284 ◽  
Author(s):  
Roy L. P. G. Jentjens ◽  
Luke Moseley ◽  
Rosemary H. Waring ◽  
Leslie K. Harding ◽  
Asker E. Jeukendrup
Keyword(s):  
Power Output ◽  
Statistical Significance ◽  
Random Order ◽  
Carbohydrate Oxidation ◽  
Maximum Power ◽  
The Other ◽  
Cycling Exercise ◽  
Oxidation Rates ◽  
Maximum Power Output ◽  
Exercise Period

The purpose of the present study was to examine whether combined ingestion of a large amount of fructose and glucose during cycling exercise would lead to exogenous carbohydrate oxidation rates >1 g/min. Eight trained cyclists (maximal O2consumption: 62 ± 3 ml·kg-1·min-1) performed four exercise trials in random order. Each trial consisted of 120 min of cycling at 50% maximum power output (63 ± 2% maximal O2consumption), while subjects received a solution providing either 1.2 g/min of glucose (Med-Glu), 1.8 g/min of glucose (High-Glu), 0.6 g/min of fructose + 1.2 g/min of glucose (Fruc+Glu), or water. The ingested fructose was labeled with [U-13C]fructose, and the ingested glucose was labeled with [U-14C]glucose. Peak exogenous carbohydrate oxidation rates were ∼55% higher ( P < 0.001) in Fruc+Glu (1.26 ± 0.07 g/min) compared with Med-Glu and High-Glu (0.80 ± 0.04 and 0.83 ± 0.05 g/min, respectively). Furthermore, the average exogenous carbohydrate oxidation rates over the 60- to 120-min exercise period were higher ( P < 0.001) in Fruc+Glu compared with Med-Glu and High-Glu (1.16 ± 0.06, 0.75 ± 0.04, and 0.75 ± 0.04 g/min, respectively). There was a trend toward a lower endogenous carbohydrate oxidation in Fruc+Glu compared with the other two carbohydrate trials, but this failed to reach statistical significance ( P = 0.075). The present results demonstrate that, when fructose and glucose are ingested simultaneously at high rates during cycling exercise, exogenous carbohydrate oxidation rates can reach peak values of ∼1.3 g/min.

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