16,865 research outputs found
Graphs that do not contain a cycle with a node that has at least two neighbors on it
We recall several known results about minimally 2-connected graphs, and show
that they all follow from a decomposition theorem. Starting from an analogy
with critically 2-connected graphs, we give structural characterizations of the
classes of graphs that do not contain as a subgraph and as an induced subgraph,
a cycle with a node that has at least two neighbors on the cycle. From these
characterizations we get polynomial time recognition algorithms for these
classes, as well as polynomial time algorithms for vertex-coloring and
edge-coloring
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
A structure theorem for graphs with no cycle with a unique chord and its consequences
We give a structural description of the class C of graphs that do not contain a cycle with a unique chord as an induced subgraph. Our main theorem states that any connected graph in C is a either in some simple basic class or has a decomposition. Basic classes are cliques, bipartite graphs with one side containing only nodes of degree two and induced subgraph of the famous Heawood or Petersen graph. Decompositions are node cutsets consisting of one or two nodes and edge cutsets called 1-joins. Our decomposition theorem actually gives a complete structure theorem for C, i.e. every graph in C can be built from basic graphs that can be explicitly constructed, and gluing them together by prescribed composition operations ; and all graphs built this way are in C. This has several consequences : an O(nm)-time algorithm to decide whether a graph is in C, an O(n+m)-time algorithm that finds a maximum clique of any graph in C and an O(nm)-time coloring algorithm for graphs in C. We prove that every graph in C is either 3-colorable or has a coloring with ω colors where ω is the size of a largest clique. The problem of finding a maximum stable set for a graph in C is known to be NP-hard.Cycle with a unique chord, decomposition, structure, detection, recognition, Heawood graph, Petersen graph, coloring.
Distributed Detection of Cycles
Distributed property testing in networks has been introduced by Brakerski and
Patt-Shamir (2011), with the objective of detecting the presence of large dense
sub-networks in a distributed manner. Recently, Censor-Hillel et al. (2016)
have shown how to detect 3-cycles in a constant number of rounds by a
distributed algorithm. In a follow up work, Fraigniaud et al. (2016) have shown
how to detect 4-cycles in a constant number of rounds as well. However, the
techniques in these latter works were shown not to generalize to larger cycles
with . In this paper, we completely settle the problem of cycle
detection, by establishing the following result. For every , there
exists a distributed property testing algorithm for -freeness, performing
in a constant number of rounds. All these results hold in the classical CONGEST
model for distributed network computing. Our algorithm is 1-sided error. Its
round-complexity is where is the property
testing parameter measuring the gap between legal and illegal instances
A simple linear time algorithm for the locally connected spanning tree problem on maximal planar chordal graphs
A locally connected spanning tree (LCST) T of a graph G is a spanning tree of G such that, for each node, its neighborhood in T induces a connected sub- graph in G. The problem of determining whether a graph contains an LCST or not has been proved to be NP-complete, even if the graph is planar or chordal. The main result of this paper is a simple linear time algorithm that, given a maximal planar chordal graph, determines in linear time whether it contains an LCST or not, and produces one if it exists. We give an anal- ogous result for the case when the input graph is a maximal outerplanar graph
Coalition Resilient Outcomes in Max k-Cut Games
We investigate strong Nash equilibria in the \emph{max -cut game}, where
we are given an undirected edge-weighted graph together with a set of colors. Nodes represent players and edges capture their mutual
interests. The strategy set of each player consists of the colors. When
players select a color they induce a -coloring or simply a coloring. Given a
coloring, the \emph{utility} (or \emph{payoff}) of a player is the sum of
the weights of the edges incident to , such that the color chosen
by is different from the one chosen by . Such games form some of the
basic payoff structures in game theory, model lots of real-world scenarios with
selfish agents and extend or are related to several fundamental classes of
games.
Very little is known about the existence of strong equilibria in max -cut
games. In this paper we make some steps forward in the comprehension of it. We
first show that improving deviations performed by minimal coalitions can cycle,
and thus answering negatively the open problem proposed in
\cite{DBLP:conf/tamc/GourvesM10}. Next, we turn our attention to unweighted
graphs. We first show that any optimal coloring is a 5-SE in this case. Then,
we introduce -local strong equilibria, namely colorings that are resilient
to deviations by coalitions such that the maximum distance between every pair
of nodes in the coalition is at most . We prove that -local strong
equilibria always exist. Finally, we show the existence of strong Nash
equilibria in several interesting specific scenarios.Comment: A preliminary version of this paper will appear in the proceedings of
the 45th International Conference on Current Trends in Theory and Practice of
Computer Science (SOFSEM'19
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