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The complexity of recognizing minimally tough graphs
A graph is called -tough if the removal of any vertex set that
disconnects the graph leaves at most components. The toughness of a
graph is the largest for which the graph is -tough. A graph is minimally
-tough if the toughness of the graph is and the deletion of any edge
from the graph decreases the toughness. The complexity class DP is the set of
all languages that can be expressed as the intersection of a language in NP and
a language in coNP. In this paper, we prove that recognizing minimally
-tough graphs is DP-complete for any positive rational number . We
introduce a new notion called weighted toughness, which has a key role in our
proof
Properties of minimally -tough graphs
A graph is minimally -tough if the toughness of is and the
deletion of any edge from decreases the toughness. Kriesell conjectured
that for every minimally -tough graph the minimum degree . We
show that in every minimally -tough graph . We
also prove that every minimally -tough claw-free graph is a cycle. On the
other hand, we show that for every any graph can be embedded
as an induced subgraph into a minimally -tough graph
Minimally toughness in special graph classes
Let be a positive real number. A graph is called -tough, if the
removal of any cutset leaves at most components. The toughness of a
graph is the largest for which the graph is -tough. A graph is minimally
-tough, if the toughness of the graph is and the deletion of any edge
from the graph decreases the toughness. In this paper we investigate the
minimum degree and the recognizability of minimally -tough graphs in the
class of chordal graphs, split graphs, claw-free graphs and -free graphs
Conditions for minimally tough graphs
Katona, Solt\'esz, and Varga showed that no induced subgraph can be excluded
from the class of minimally tough graphs. In this paper, we consider the
opposite question, namely which induced subgraphs, if any, must necessarily be
present in each minimally -tough graph.
Katona and Varga showed that for any rational number , every
minimally -tough graph contains a hole. We complement this result by showing
that for any rational number , every minimally -tough graph must
contain either a hole or an induced subgraph isomorphic to the -sun for some
integer .
We also show that for any rational number , every minimally
-tough graph must contain either an induced -cycle, an induced -cycle,
or two independent edges as an induced subgraph
Conditions for minimally tough graphs
Katona, Soltész, and Varga showed that no induced subgraph can be excluded from the class of minimally tough graphs. In this paper, we consider the opposite question, namely which induced subgraphs, if any, must necessarily be present in each minimally t-tough graph.
Katona and Varga showed that for any rational number t∈(1/2,1], every minimally t-tough graph contains a hole. We complement this result by showing that for any rational number t>1, every minimally t-tough graph must contain either a hole or an induced subgraph isomorphic to the k-sun for some integer k≥3.
We also show that for any rational number t>1/2, every minimally t-tough graph must contain either an induced 4-cycle, an induced 5-cycle, or two independent edges as an induced subgraph
Higher Education Exchange: 2009
This annual publication serves as a forum for new ideas and dialogue between scholars and the larger public. Essays explore ways that students, administrators, and faculty can initiate and sustain an ongoing conversation about the public life they share.The Higher Education Exchange is founded on a thought articulated by Thomas Jefferson in 1820: "I know no safe depository of the ultimate powers of the society but the people themselves; and if we think them not enlightened enough to exercise their control with a wholesome discretion, the remedy is not to take it from them, but to inform their discretion by education."In the tradition of Jefferson, the Higher Education Exchange agrees that a central goal of higher education is to help make democracy possible by preparing citizens for public life. The Higher Education Exchange is part of a movement to strengthen higher education's democratic mission and foster a more democratic culture throughout American society.Working in this tradition, the Higher Education Exchange publishes interviews, case studies, analyses, news, and ideas about efforts within higher education to develop more democratic societies
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