2,050 research outputs found
(Total) Vector Domination for Graphs with Bounded Branchwidth
Given a graph of order and an -dimensional non-negative
vector , called demand vector, the vector domination
(resp., total vector domination) is the problem of finding a minimum
such that every vertex in (resp., in ) has
at least neighbors in . The (total) vector domination is a
generalization of many dominating set type problems, e.g., the dominating set
problem, the -tuple dominating set problem (this is different from the
solution size), and so on, and its approximability and inapproximability have
been studied under this general framework. In this paper, we show that a
(total) vector domination of graphs with bounded branchwidth can be solved in
polynomial time. This implies that the problem is polynomially solvable also
for graphs with bounded treewidth. Consequently, the (total) vector domination
problem for a planar graph is subexponential fixed-parameter tractable with
respectto , where is the size of solution.Comment: 16 page
The Price of Connectivity for Vertex Cover
The vertex cover number of a graph is the minimum number of vertices that are
needed to cover all edges. When those vertices are further required to induce a
connected subgraph, the corresponding number is called the connected vertex
cover number, and is always greater or equal to the vertex cover number.
Connected vertex covers are found in many applications, and the relationship
between those two graph invariants is therefore a natural question to
investigate. For that purpose, we introduce the {\em Price of Connectivity},
defined as the ratio between the two vertex cover numbers. We prove that the
price of connectivity is at most 2 for arbitrary graphs. We further consider
graph classes in which the price of connectivity of every induced subgraph is
bounded by some real number . We obtain forbidden induced subgraph
characterizations for every real value .
We also investigate critical graphs for this property, namely, graphs whose
price of connectivity is strictly greater than that of any proper induced
subgraph. Those are the only graphs that can appear in a forbidden subgraph
characterization for the hereditary property of having a price of connectivity
at most . In particular, we completely characterize the critical graphs that
are also chordal.
Finally, we also consider the question of computing the price of connectivity
of a given graph. Unsurprisingly, the decision version of this question is
NP-hard. In fact, we show that it is even complete for the class , the class of decision problems that can be solved in polynomial
time, provided we can make queries to an NP-oracle. This paves the
way for a thorough investigation of the complexity of problems involving ratios
of graph invariants.Comment: 19 pages, 8 figure
Packing 3-vertex paths in claw-free graphs and related topics
An L-factor of a graph G is a spanning subgraph of G whose every component is
a 3-vertex path. Let v(G) be the number of vertices of G and d(G) the
domination number of G. A claw is a graph with four vertices and three edges
incident to the same vertex. A graph is claw-free if it has no induced subgraph
isomorphic to a claw. Our results include the following. Let G be a 3-connected
claw-free graph, x a vertex in G, e = xy an edge in G, and P a 3-vertex path in
G. Then
(a1) if v(G) = 0 mod 3, then G has an L-factor containing (avoiding) e, (a2)
if v(G) = 1 mod 3, then G - x has an L-factor, (a3) if v(G) = 2 mod 3, then G -
{x,y} has an L-factor, (a4) if v(G) = 0 mod 3 and G is either cubic or
4-connected, then G - P has an L-factor, (a5) if G is cubic with v(G) > 5 and E
is a set of three edges in G, then G - E has an L-factor if and only if the
subgraph induced by E in G is not a claw and not a triangle, (a6) if v(G) = 1
mod 3, then G - {v,e} has an L-factor for every vertex v and every edge e in G,
(a7) if v(G) = 1 mod 3, then there exist a 4-vertex path N and a claw Y in G
such that G - N and G - Y have L-factors, and (a8) d(G) < v(G)/3 +1 and if in
addition G is not a cycle and v(G) = 1 mod 3, then d(G) < v(G)/3.
We explore the relations between packing problems of a graph and its line
graph to obtain some results on different types of packings. We also discuss
relations between L-packing and domination problems as well as between induced
L-packings and the Hadwiger conjecture.
Keywords: claw-free graph, cubic graph, vertex disjoint packing, edge
disjoint packing, 3-vertex factor, 3-vertex packing, path-factor, induced
packing, graph domination, graph minor, the Hadwiger conjecture.Comment: 29 page
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