85 research outputs found
The Geodetic Hull Number is Hard for Chordal Graphs
We show the hardness of the geodetic hull number for chordal graphs
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
On the Steiner, geodetic and hull numbers of graphs
Given a graph G and a subset W ? V (G), a Steiner W-tree is a tree of minimum
order that contains all of W. Let S(W) denote the set of all vertices in G that lie on
some Steiner W-tree; we call S(W) the Steiner interval of W. If S(W) = V (G), then
we call W a Steiner set of G. The minimum order of a Steiner set of G is called the
Steiner number of G.
Given two vertices u, v in G, a shortest u − v path in G is called a u − v geodesic.
Let I[u, v] denote the set of all vertices in G lying on some u − v geodesic, and let
J[u, v] denote the set of all vertices in G lying on some induced u − v path. Given a
set S ? V (G), let I[S] = ?u,v?S I[u, v], and let J[S] = ?u,v?S J[u, v]. We call I[S]
the geodetic closure of S and J[S] the monophonic closure of S. If I[S] = V (G), then
S is called a geodetic set of G. If J[S] = V (G), then S is called a monophonic set of
G. The minimum order of a geodetic set in G is named the geodetic number of G.
In this paper, we explore the relationships both between Steiner sets and geodetic
sets and between Steiner sets and monophonic sets. We thoroughly study the relationship
between the Steiner number and the geodetic number, and address the questions:
in a graph G when must every Steiner set also be geodetic and when must every Steiner
set also be monophonic. In particular, among others we show that every Steiner set
in a connected graph G must also be monophonic, and that every Steiner set in a
connected interval graph H must be geodetic
On geodesic and monophonic convexity
In this paper we deal with two types of graph convexities, which are the most natural path convexities in a graph and which are defined by a system P of paths in a connected graph G: the geodesic convexity (also called metric
convexity) which arises when we consider shortest paths, and the monophonic convexity (also called minimal path convexity) when we consider chordless paths. First, we present a realization theorem proving, that there is no general relationship between monophonic and geodetic hull sets. Second, we study the contour of a graph, showing that the contour must be monophonic. Finally, we consider the so-called edge Steiner sets. We prove that every edge Steiner set is edge monophonic.Ministerio de Ciencia y TecnologÃaFondo Europeo de Desarrollo RegionalGeneralitat de Cataluny
Rebuilding convex sets in graphs
The usual distance between pairs of vertices in a graph naturally gives rise to the notion of an interval between a pair of vertices in a graph. This in turn allows us to extend the notions of convex sets, convex hull, and extreme points in Euclidean space to the vertex set of a graph. The extreme vertices of a graph are known to be precisely the simplicial vertices, i.e., the vertices whose neighborhoods are complete graphs. It is known that the class of graphs with the Minkowski–Krein–Milman property, i.e., the property that every convex set is the convex hull of its extreme points, is precisely the class of chordal graphs without induced 3-fans. We define a vertex to be a contour vertex if the eccentricity of every neighbor is at most as large as that of the vertex. In this paper we show that every convex set of vertices in a graph is the convex hull of the collection of its contour vertices. We characterize those graphs for which every convex set has the property that its contour vertices coincide with its extreme points. A set of vertices in a graph is a geodetic set if the union of the intervals between pairs of vertices in the set, taken over all pairs in the set, is the entire vertex set. We show that the contour vertices in distance hereditary graphs form a geodetic set
On the Computational Complexity of the Strong Geodetic Recognition Problem
A strong geodetic set of a graph~ is a vertex set~
in which it is possible to cover all the remaining vertices of~ by assigning a unique shortest path between each vertex pair of~. In the
Strong Geodetic problem (SG) a graph~ and a positive integer~ are given
as input and one has to decide whether~ has a strong geodetic set of
cardinality at most~. This problem is known to be NP-hard for general
graphs. In this work we introduce the Strong Geodetic Recognition problem
(SGR), which consists in determining whether even a given vertex set~ is strong geodetic. We demonstrate that this version is
NP-complete. We investigate and compare the computational complexity of both
decision problems restricted to some graph classes, deriving polynomial-time
algorithms, NP-completeness proofs, and initial parameterized complexity
results, including an answer to an open question in the literature for the
complexity of SG for chordal graphs
Parameterized Complexity of Geodetic Set
A vertex set S of a graph G is geodetic if every vertex of G lies on a shortest path between two vertices in S. Given a graph G and k ? ?, the NP-hard Geodetic Set problem asks whether there is a geodetic set of size at most k. Complementing various works on Geodetic Set restricted to special graph classes, we initiate a parameterized complexity study of Geodetic Set and show, on the negative side, that Geodetic Set is W[1]-hard when parameterized by feedback vertex number, path-width, and solution size, combined. On the positive side, we develop fixed-parameter algorithms with respect to the feedback edge number, the tree-depth, and the modular-width of the input graph
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