31 research outputs found
Convex Independence in Permutation Graphs
A set C of vertices of a graph is P_3-convex if every vertex outside C has at
most one neighbor in C. The convex hull \sigma(A) of a set A is the smallest
P_3-convex set that contains A. A set M is convexly independent if for every
vertex x \in M, x \notin \sigma(M-x). We show that the maximal number of
vertices that a convexly independent set in a permutation graph can have, can
be computed in polynomial time
The Detour Monophonic Convexity Number of a Graph
A set Ā is detour monophonic convexif Ā The detour monophonic convexity number is denoted by Ā is the cardinality of a maximum proper detour monophonic convex subset ofĀ Some general properties satisfied by this concept are studied. The detour monophonic convexity number of certain classes of graphs are determined. It is shown that for every pair of integersĀ Ā and Ā with Ā there exists a connected graph Ā such thatĀ Ā and , where Ā is the monophonic convexity number of
Convexities related to path properties on graphs; a unified approach
Path properties, such as 'geodesic', 'induced', 'all paths' define a convexity on a connected graph. The general notion of path property, introduced in this paper, gives rise to a comprehensive survey of results obtained by different authors for a variety of path properties, together with a number of new results. We pay special attention to convexities defined by path properties on graph products and the classical convexity invariants, such as the Caratheodory, Helly and Radon numbers in relation with graph invariants, such as clique numbers and other graph properties.
The induced path function, monotonicity and betweenness
The induced path function of a graph consists of the set of all vertices lying on the induced paths between vertices and . This function is a special instance of a transit function. The function satisfies betweenness if implies and implies , and it is monotone if implies . The induced path function of aconnected graph satisfying the betweenness and monotone axioms are characterized by transit axioms.betweenness;induced path;transit function;monotone;house domino;long cycle;p-graph
On the hull and interval numbers of oriented graphs
In this work, for a given oriented graph , we study its interval and hull
numbers, denoted by and , respectively, in the geodetic,
and convexities. This last one, we believe to be formally
defined and first studied in this paper, although its undirected version is
well-known in the literature. Concerning bounds, for a strongly oriented graph
, we prove that and that there is a strongly
oriented graph such that . We also determine exact
values for the hull numbers in these three convexities for tournaments, which
imply polynomial-time algorithms to compute them. These results allows us to
deduce polynomial-time algorithms to compute when the
underlying graph of is split or cobipartite. Moreover, we provide a
meta-theorem by proving that if deciding whether or
is NP-hard or W[i]-hard parameterized by , for some
, then the same holds even if the underlying graph of
is bipartite. Next, we prove that deciding whether or
is W[2]-hard parameterized by , even if the
underlying graph of is bipartite; that deciding whether or is NP-complete, even if has no directed
cycles and the underlying graph of is a chordal bipartite graph; and that
deciding whether or is W[2]-hard
parameterized by , even if the underlying graph of is split. We also
argue that the interval and hull numbers in the oriented and
convexities can be computed in polynomial time for graphs of bounded tree-width
by using Courcelle's theorem
Byzantine Approximate Agreement on Graphs
Consider a distributed system with n processors out of which f can be Byzantine faulty. In the approximate agreement task, each processor i receives an input value x_i and has to decide on an output value y_i such that
1) the output values are in the convex hull of the non-faulty processors\u27 input values,
2) the output values are within distance d of each other.
Classically, the values are assumed to be from an m-dimensional Euclidean space, where m >= 1.
In this work, we study the task in a discrete setting, where input values with some structure expressible as a graph. Namely, the input values are vertices of a finite graph G and the goal is to output vertices that are within distance d of each other in G, but still remain in the graph-induced convex hull of the input values. For d=0, the task reduces to consensus and cannot be solved with a deterministic algorithm in an asynchronous system even with a single crash fault. For any d >= 1, we show that the task is solvable in asynchronous systems when G is chordal and n > (omega+1)f, where omega is the clique number of G. In addition, we give the first Byzantine-tolerant algorithm for a variant of lattice agreement. For synchronous systems, we show tight resilience bounds for the exact variants of these and related tasks over a large class of combinatorial structures
Transit functions on graphs (and posets)
The notion of transit function is introduced to present a unifying approachfor results and ideas on intervals, convexities and betweenness in graphs andposets. Prime examples of such transit functions are the interval function I andthe induced path function J of a connected graph. Another transit function isthe all-paths function. New transit functions are introduced, such as the cutvertextransit function and the longest path function. The main idea of transitfunctions is that of Ć¢ā¬ĖtransferringĆ¢ā¬ā¢ problems and ideas of one transit functionto the other. For instance, a result on the interval function I might suggestsimilar problems for the induced path function J. Examples are given of howfruitful this transfer can be. A list of Prototype Problems and Questions forthis transferring process is given, which suggests many new questions and openproblems.graph theory;betweenness;block graph;convexity;distance in graphs;interval function;path function;induced path;paths and cycles;transit function;types of graphs
The general position number and the iteration time in the P3 convexity
In this paper, we investigate two graph convexity parameters: the iteration
time and the general position number. Harary and Nieminem introduced in 1981
the iteration time in the geodesic convexity, but its computational complexity
was still open. Manuel and Klav\v{z}ar introduced in 2018 the general position
number of the geodesic convexity and proved that it is NP-hard to compute. In
this paper, we extend these parameters to the P3 convexity and prove that it is
NP-hard to compute them. With this, we also prove that the iteration number is
NP-hard on the geodesic convexity even in graphs with diameter two. These
results are the last three missing NP-hardness results regarding the ten most
studied graph convexity parameters in the geodesic and P3 convexities