15,650 research outputs found
Bounds on the radius and status of graphs
Two classical concepts of centrality in a graph are the median and the
center. The connected notions of the status and the radius of a graph seem to
be in no relation. In this paper, however, we show a clear connection of both
concepts, as they obtain their minimum and maximum values at the same type of
tree graphs. Trees with fixed maximum degree and extremum radius and status,
resp., are characterized. The bounds on radius and status can be transferred to
general connected graphs via spanning trees.
A new method of proof allows not only to regain results of Lin et al. on
graphs with extremum status, but it allows also to prove analogous results on
graphs with extremum radius
Algorithms and almost tight results for 3-colorability of small diameter graphs.
The 3-coloring problem is well known to be NP-complete. It is also well known that it remains NP-complete when the input is restricted to graphs with diameter 4. Moreover, assuming the Exponential Time Hypothesis (ETH), 3-coloring cannot be solved in time 2o(n) on graphs with n vertices and diameter at most 4. In spite of extensive studies of the 3-coloring problem with respect to several basic parameters, the complexity status of this problem on graphs with small diameter, i.e. with diameter at most 2, or at most 3, has been an open problem. In this paper we investigate graphs with small diameter. For graphs with diameter at most 2, we provide the first subexponential algorithm for 3-coloring, with complexity 2O(nlogn√). Furthermore we extend the notion of an articulation vertex to that of an articulation neighborhood, and we provide a polynomial algorithm for 3-coloring on graphs with diameter 2 that have at least one articulation neighborhood. For graphs with diameter at most 3, we establish the complexity of 3-coloring by proving for every ε∈[0,1) that 3-coloring is NP-complete on triangle-free graphs of diameter 3 and radius 2 with n vertices and minimum degree δ=Θ(nε). Moreover, assuming ETH, we use three different amplification techniques of our hardness results, in order to obtain for every ε∈[0,1) subexponential asymptotic lower bounds for the complexity of 3-coloring on triangle-free graphs with diameter 3 and minimum degree δ=Θ(nε). Finally, we provide a 3-coloring algorithm with running time 2O(min{δΔ, nδlogδ}) for arbitrary graphs with diameter 3, where n is the number of vertices and δ (resp. Δ) is the minimum (resp. maximum) degree of the input graph. To the best of our knowledge, this is the first subexponential algorithm for graphs with δ=ω(1) and for graphs with δ=O(1) and Δ=o(n). Due to the above lower bounds of the complexity of 3-coloring, the running time of this algorithm is asymptotically almost tight when the minimum degree of the input graph is δ=Θ(nε), where ε∈[12,1), as its time complexity is 2O(nδlogδ)=2O(n1−εlogn) and the corresponding lower bound states that there is no 2o(n1−ε)-time algorithm
The Max-Distance Network Creation Game on General Host Graphs
In this paper we study a generalization of the classic \emph{network creation
game} in the scenario in which the players sit on a given arbitrary
\emph{host graph}, which constrains the set of edges a player can activate at a
cost of each. This finds its motivations in the physical
limitations one can have in constructing links in practice, and it has been
studied in the past only when the routing cost component of a player is given
by the sum of distances to all the other nodes. Here, we focus on another
popular routing cost, namely that which takes into account for each player its
\emph{maximum} distance to any other player. For this version of the game, we
first analyze some of its computational and dynamic aspects, and then we
address the problem of understanding the structure of associated pure Nash
equilibria. In this respect, we show that the corresponding price of anarchy
(PoA) is fairly bad, even for several basic classes of host graphs. More
precisely, we first exhibit a lower bound of
for any . Notice that this implies a counter-intuitive lower
bound of for very small values of (i.e., edges can
be activated almost for free). Then, we show that when the host graph is
restricted to be either -regular (for any constant ), or a
2-dimensional grid, the PoA is still , which is proven to be tight for
. On the positive side, if , we show
the PoA is . Finally, in the case in which the host graph is very sparse
(i.e., , with ), we prove that the PoA is , for any
.Comment: 17 pages, 4 figure
Fast Distributed Computation of Distances in Networks
This paper presents a distributed algorithm to simultaneously compute the
diameter, radius and node eccentricity in all nodes of a synchronous network.
Such topological information may be useful as input to configure other
algorithms. Previous approaches have been modular, progressing in sequential
phases using building blocks such as BFS tree construction, thus incurring
longer executions than strictly required. We present an algorithm that, by
timely propagation of available estimations, achieves a faster convergence to
the correct values. We show local criteria for detecting convergence in each
node. The algorithm avoids the creation of BFS trees and simply manipulates
sets of node ids and hop counts. For the worst scenario of variable start
times, each node i with eccentricity ecc(i) can compute: the node eccentricity
in diam(G)+ecc(i)+2 rounds; the diameter in 2*diam(G)+ecc(i)+2 rounds; and the
radius in diam(G)+ecc(i)+2*radius(G) rounds.Comment: 12 page
Distinguishing Infections on Different Graph Topologies
The history of infections and epidemics holds famous examples where
understanding, containing and ultimately treating an outbreak began with
understanding its mode of spread. Influenza, HIV and most computer viruses,
spread person to person, device to device, through contact networks; Cholera,
Cancer, and seasonal allergies, on the other hand, do not. In this paper we
study two fundamental questions of detection: first, given a snapshot view of a
(perhaps vanishingly small) fraction of those infected, under what conditions
is an epidemic spreading via contact (e.g., Influenza), distinguishable from a
"random illness" operating independently of any contact network (e.g., seasonal
allergies); second, if we do have an epidemic, under what conditions is it
possible to determine which network of interactions is the main cause of the
spread -- the causative network -- without any knowledge of the epidemic, other
than the identity of a minuscule subsample of infected nodes?
The core, therefore, of this paper, is to obtain an understanding of the
diagnostic power of network information. We derive sufficient conditions
networks must satisfy for these problems to be identifiable, and produce
efficient, highly scalable algorithms that solve these problems. We show that
the identifiability condition we give is fairly mild, and in particular, is
satisfied by two common graph topologies: the grid, and the Erdos-Renyi graphs
- …