1,093 research outputs found
Feedback vertex set on chordal bipartite graphs
Let G=(A,B,E) be a bipartite graph with color classes A and B. The graph G is
chordal bipartite if G has no induced cycle of length more than four. Let
G=(V,E) be a graph. A feedback vertex set F is a set of vertices F subset V
such that G-F is a forest. The feedback vertex set problem asks for a feedback
vertex set of minimal cardinality. We show that the feedback vertex set problem
can be solved in polynomial time on chordal bipartite graphs
Solving problems on generalized convex graphs via mim-width
A bipartite graph G = (A, B, E) is H-convex, for some family of graphs H, if there exists a graph H β H with V (H) = A such that the set of neighbours in A of each b β B induces a connected subgraph of H. Many NP-complete problems become polynomial-time solvable for H-convex graphs when H is the set of paths. In this case, the class of H-convex graphs is known as the class of convex graphs. The underlying reason is that this class has bounded mim-width. We extend the latter result to families of H-convex graphs where (i) H is the set of cycles, or (ii) H is the set of trees with bounded maximum degree and a bounded number of vertices of degree at least 3. As a consequence, we can reprove and strengthen a large number of results on generalized convex graphs known in the literature. To complement result (ii), we show that the mim-width of H-convex graphs is unbounded if H is the set of trees with arbitrarily large maximum degree or an arbitrarily large number of vertices of degree at least 3. In this way we are able to determine complexity dichotomies for the aforementioned graph problems. Afterwards we perform a more refined width-parameter analysis, which shows even more clearly which width parameters are bounded for classes of H-convex graphs
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
Algorithmic Aspects of Secure Connected Domination in Graphs
Let be a simple, undirected and connected graph. A connected
dominating set is a secure connected dominating set of , if
for each , there exists such that
and the set is a connected dominating set
of . The minimum size of a secure connected dominating set of denoted by
, is called the secure connected domination number of .
Given a graph and a positive integer the Secure Connected Domination
(SCDM) problem is to check whether has a secure connected dominating set
of size at most In this paper, we prove that the SCDM problem is
NP-complete for doubly chordal graphs, a subclass of chordal graphs. We
investigate the complexity of this problem for some subclasses of bipartite
graphs namely, star convex bipartite, comb convex bipartite, chordal bipartite
and chain graphs. The Minimum Secure Connected Dominating Set (MSCDS) problem
is to find a secure connected dominating set of minimum size in the input
graph. We propose a - approximation algorithm for MSCDS,
where is the maximum degree of the input graph and prove
that MSCDS cannot be approximated within for any unless even for
bipartite graphs. Finally, we show that the MSCDS is APX-complete for graphs
with
- β¦