34 research outputs found
On non-traceable, non-hypotraceable, arachnoid graphs
Motivated by questions concerning optical networks, in 2003 Gargano, Hammar, Hell, Stacho, and Vaccaro defined the notions of spanning spiders and arachnoid graphs. A spider is a tree with at most one branch (vertex of degree at least 3). The spider is centred at the branch vertex (if there is any,otherwise it is centred at any of the vertices). A graph is arachnoid if it has
a spanning spider centred at any of its vertices. Traceable graphs are obviously arachnoid, and Gargano et al. observed that hypotraceable graphs (non-traceable graphs with the property that all vertex-deleted subgraphs are
traceable) are also easily seen to be arachnoid. However, they did not find any other arachnoid graphs, and asked the question whether they exist. The main goal of this paper is to answer this question in the affirmative, moreover, we show that for any prescribed graph H, there exists a non-traceable, non-hypotraceable, arachnoid graph that contains H as an induced subgraph
Search for the end of a path in the d-dimensional grid and in other graphs
We consider the worst-case query complexity of some variants of certain
\cl{PPAD}-complete search problems. Suppose we are given a graph and a
vertex . We denote the directed graph obtained from by
directing all edges in both directions by . is a directed subgraph of
which is unknown to us, except that it consists of vertex-disjoint
directed paths and cycles and one of the paths originates in . Our goal is
to find an endvertex of a path by using as few queries as possible. A query
specifies a vertex , and the answer is the set of the edges of
incident to , together with their directions. We also show lower bounds for
the special case when consists of a single path. Our proofs use the theory
of graph separators. Finally, we consider the case when the graph is a grid
graph. In this case, using the connection with separators, we give
asymptotically tight bounds as a function of the size of the grid, if the
dimension of the grid is considered as fixed. In order to do this, we prove a
separator theorem about grid graphs, which is interesting on its own right
Finding a non-minority ball with majority answers
Suppose we are given a set of balls each colored
either red or blue in some way unknown to us. To find out some information
about the colors, we can query any triple of balls
. As an answer to such a query we obtain (the
index of) a {\em majority ball}, that is, a ball whose color is the same as the
color of another ball from the triple. Our goal is to find a {\em non-minority
ball}, that is, a ball whose color occurs at least times among the
balls. We show that the minimum number of queries needed to solve this
problem is in the adaptive case and in the
non-adaptive case. We also consider some related problems
Spiders everywhere
A spider is a tree with at most one branch (a vertex of degree at least 3) centred at the branch if it exists, and centred at any vertex otherwise. A graph G is arachnoid if for any vertex v of G, there exists a spanning spider of G centred at v-in other words: there are spiders everywhere! Hypotraceable graphs are non-traceable graphs in which all vertex-deleted subgraphs are traceable. Gargano et al. (2004) defined arachnoid graphs as natural generalisations of traceable graphs and asked for the existence of arachnoid graphs that are (i) non-traceable and non-hypotraceable, or (ii) in which some vertex is the centre of only spiders with more than three legs. An affirmative answer to (ii) implies an affirmative answer to (i). While non-traceable, non-hypotraceable arachnoid graphs were described in Wiener (2017), (ii) remained open. In this paper we give an affirmative answer to this question and discuss spanning spiders whose legs must have some minimum length. (C) 2020 The Author(s). Published by Elsevier B.V
On constructions of hypotraceable graphs
A graph G is hypohamiltonian/hypotraceable if it is not hamiltonian/traceable,but all vertex deleted subgraphs of G are hamiltonian/traceable. Until now all hypotraceable graphs were constructed using hypohamiltonian graphs; extending a method of Thomassen we present a construction that uses so-called almost hypohamiltonian graphs (nonhamiltonian graphs, whose vertex deleted subgraphs
are hamiltonian with exactly one exception). As an application, we construct a planar hypotraceable graph of order 138, improving the best known bound of 154. We also prove a structural type theorem showing that hypotraceable graphs possessing some connectivity properties are all built using either Thomassen's or our method