53 research outputs found
Connectedness of the graph of vertex-colourings
For a positive integer k and a graph G, the k-colour graph of G , Ck(G), is the graph that has the proper k-vertex-colourings of G as its vertex set, and two k -colourings are joined by an edge in Ck(G) if they differ in colour on just one vertex of G . In this note some results on the connectedness of Ck(G) are proved. In particular, it is shown that if G has chromatic number k∈{2,3}, then Ck(G) is not connected. On the other hand, for k⩾4 there are graphs with chromatic number k for which Ck(G) is not connected, and there are k -chromatic graphs for which Ck(G) is connected
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
Reconfiguration of Dominating Sets
We explore a reconfiguration version of the dominating set problem, where a
dominating set in a graph is a set of vertices such that each vertex is
either in or has a neighbour in . In a reconfiguration problem, the goal
is to determine whether there exists a sequence of feasible solutions
connecting given feasible solutions and such that each pair of
consecutive solutions is adjacent according to a specified adjacency relation.
Two dominating sets are adjacent if one can be formed from the other by the
addition or deletion of a single vertex.
For various values of , we consider properties of , the graph
consisting of a vertex for each dominating set of size at most and edges
specified by the adjacency relation. Addressing an open question posed by Haas
and Seyffarth, we demonstrate that is not necessarily
connected, for the maximum cardinality of a minimal dominating set
in . The result holds even when graphs are constrained to be planar, of
bounded tree-width, or -partite for . Moreover, we construct an
infinite family of graphs such that has exponential
diameter, for the minimum size of a dominating set. On the positive
side, we show that is connected and of linear diameter for any
graph on vertices having at least independent edges.Comment: 12 pages, 4 figure
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