14 research outputs found
Upper and lower bounds for finding connected motifs in vertex-colored graphs
International audienceWe study the problem of finding occurrences of motifs in vertex-colored graphs, where a motif is a multiset of colors, and an occurrence of a motif is a subset of connected vertices whose multiset of colors equals the motif. This problem is a natural graph-theoretic pattern matching variant where we are not interested in the actual structure of the occurrence of the pattern, we only require it to preserve the very basic topological requirement of connectedness. We give two positive results and three negative results that together give an extensive picture of tractable and intractable instances of the problem
Using Neighborhood Diversity to Solve Hard Problems
Parameterized algorithms are a very useful tool for dealing with NP-hard
problems on graphs. Yet, to properly utilize parameterized algorithms it is
necessary to choose the right parameter based on the type of problem and
properties of the target graph class. Tree-width is an example of a very
successful graph parameter, however it cannot be used on dense graph classes
and there also exist problems which are hard even on graphs of bounded
tree-width. Such problems can be tackled by using vertex cover as a parameter,
however this places severe restrictions on admissible graph classes.
Michael Lampis has recently introduced neighborhood diversity, a new graph
parameter which generalizes vertex cover to dense graphs. Among other results,
he has shown that simple parameterized algorithms exist for a few problems on
graphs of bounded neighborhood diversity. Our article further studies this area
and provides new algorithms parameterized by neighborhood diversity for the
p-Vertex-Disjoint Paths, Graph Motif and Precoloring Extension problems -- the
latter two being hard even on graphs of bounded tree-width
Tropical Dominating Sets in Vertex-Coloured Graphs
Given a vertex-coloured graph, a dominating set is said to be tropical if
every colour of the graph appears at least once in the set. Here, we study
minimum tropical dominating sets from structural and algorithmic points of
view. First, we prove that the tropical dominating set problem is NP-complete
even when restricted to a simple path. Then, we establish upper bounds related
to various parameters of the graph such as minimum degree and number of edges.
We also give upper bounds for random graphs. Last, we give approximability and
inapproximability results for general and restricted classes of graphs, and
establish a FPT algorithm for interval graphs.Comment: 19 pages, 4 figure
The parameterised complexity of counting connected subgraphs and graph motifs
We introduce a family of parameterised counting problems on graphs, p-#Induced Subgraph With Property(Φ), which generalises a number of problems which have previously been studied. This paper focuses on the case in which Φ defines a family of graphs whose edge-minimal elements all have bounded treewidth; this includes the special case in which Φ describes the property of being connected. We show that exactly counting the number of connected induced k-vertex subgraphs in an n-vertex graph is #W[1]-hard, but on the other hand there exists an FPTRAS for the problem; more generally, we show that there exists an FPTRAS for p-#Induced Subgraph With Property(Φ) whenever Φ is monotone and all the minimal graphs satisfying Φ have bounded treewidth. We then apply these results to a counting version of the Graph Motif problem
Balanced Connected Subgraph Problem in Geometric Intersection Graphs
We study the Balanced Connected Subgraph(shortly, BCS) problem on geometric
intersection graphs such as interval, circular-arc, permutation, unit-disk,
outer-string graphs, etc. Given a vertex-colored graph , where each
vertex in is colored with either ``red'' or ``blue'', the BCS problem seeks
a maximum cardinality induced connected subgraph of such that is
color-balanced, i.e., contains an equal number of red and blue vertices. We
study the computational complexity landscape of the BCS problem while
considering geometric intersection graphs. On one hand, we prove that the BCS
problem is NP-hard on the unit disk, outer-string, complete grid, and unit
square graphs. On the other hand, we design polynomial-time algorithms for the
BCS problem on interval, circular-arc and permutation graphs. In particular, we
give algorithm for the Steiner Tree problem on both the interval graphs and
circular arc graphs, that is used as a subroutine for solving BCS problem on
same graph classes. Finally, we present a FPT algorithm for the BCS problem on
general graphs.Comment: 17 pages, 3 figure