3 research outputs found
The -Dominating Graph
Given a graph , the -dominating graph of , , is defined to
be the graph whose vertices correspond to the dominating sets of that have
cardinality at most . Two vertices in are adjacent if and only if
the corresponding dominating sets of differ by either adding or deleting a
single vertex. The graph aids in studying the reconfiguration problem
for dominating sets. In particular, one dominating set can be reconfigured to
another by a sequence of single vertex additions and deletions, such that the
intermediate set of vertices at each step is a dominating set if and only if
they are in the same connected component of . In this paper we give
conditions that ensure is connected.Comment: 2 figure, The final publication is available at
http://link.springer.co
The Irreducible Spine(s) of Undirected Networks
Using closure concepts, we show that within every undirected network, or
graph, there is a unique irreducible subgraph which we call its "spine". The
chordless cycles which comprise this irreducible core effectively characterize
the connectivity structure of the network as a whole. In particular, it is
shown that the center of the network, whether defined by distance or
betweenness centrality, is effectively contained in this spine. By counting the
number of cycles of length 3 <= k <= max_length, we can also create a kind of
signature that can be used to identify the network. Performance is analyzed,
and the concepts we develop are illurstrated by means of a relatively small
running sample network of about 400 nodes.Comment: Submitted to WISE 201
On αrγs(k)-perfect graphs
AbstractFor some integer k⩾0 and two graph parameters π and τ, a graph G is called πτ(k)-perfect, if π(H)−τ(H)⩽k for every induced subgraph H of G. For r⩾1 let αr and γr denote the r-(distance)-independence and r-(distance)-domination number, respectively. In (J. Graph Theory 32 (1999) 303–310), I. Zverovich gave an ingenious complete characterization of α1γ1(k)-perfect graphs in terms of forbidden induced subgraphs. In this paper we study αrγs(k)-perfect graphs for r,s⩾1. We prove several properties of minimal αrγs(k)-imperfect graphs. Generalizing Zverovich's main result in (J. Graph Theory 32 (1999) 303–310), we completely characterize α2r−1γr(k)-perfect graphs for r⩾1. Furthermore, we characterize claw-free α2γ2(k)-perfect graphs