19 research outputs found
Distance domination, guarding and vertex cover for maximal outerplanar graphs
In this paper we de ne a distance guarding concept on plane graphs and associate this concept with distance domination and distance vertex cover concepts on triangulation graphs. Furthermore, for any n-vertex maximal outerplanar graph, we provide tight upper bounds for g2d(n) (2d-guarding number), γ2d(n) (2d-domination number) and β2d(n) (2d-vertex cover number).European Science FoundationMinisterio de Ciencia e InnovaciónFundação para a Ciência e a TecnologiaFondo Europeo de Desarrollo Regiona
Monitoring maximal outerplanar graphs
In this paper we define a new concept of monitoring the elements of triangulation
graphs by faces. Furthermore, we analyze this, and other monitoring concepts (by
vertices and by edges), from a combinatorial point of view, on maximal outerplanar
graphs
Conflict-Free Coloring of Planar Graphs
A conflict-free k-coloring of a graph assigns one of k different colors to
some of the vertices such that, for every vertex v, there is a color that is
assigned to exactly one vertex among v and v's neighbors. Such colorings have
applications in wireless networking, robotics, and geometry, and are
well-studied in graph theory. Here we study the natural problem of the
conflict-free chromatic number chi_CF(G) (the smallest k for which
conflict-free k-colorings exist). We provide results both for closed
neighborhoods N[v], for which a vertex v is a member of its neighborhood, and
for open neighborhoods N(v), for which vertex v is not a member of its
neighborhood.
For closed neighborhoods, we prove the conflict-free variant of the famous
Hadwiger Conjecture: If an arbitrary graph G does not contain K_{k+1} as a
minor, then chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case
bound: three colors are sometimes necessary and always sufficient. We also give
a complete characterization of the computational complexity of conflict-free
coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G,
but polynomial for outerplanar graphs. Furthermore, deciding whether
chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for
outerplanar graphs. For the bicriteria problem of minimizing the number of
colored vertices subject to a given bound k on the number of colors, we give a
full algorithmic characterization in terms of complexity and approximation for
outerplanar and planar graphs.
For open neighborhoods, we show that every planar bipartite graph has a
conflict-free coloring with at most four colors; on the other hand, we prove
that for k in {1,2,3}, it is NP-complete to decide whether a planar bipartite
graph has a conflict-free k-coloring. Moreover, we establish that any general}
planar graph has a conflict-free coloring with at most eight colors.Comment: 30 pages, 17 figures; full version (to appear in SIAM Journal on
Discrete Mathematics) of extended abstract that appears in Proceeedings of
the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA
2017), pp. 1951-196
Distance domination, guarding and covering of maximal outerplanar graphs
In this paper we introduce the notion of distance k-guarding applied to triangulation
graphs, and associate it with distance k-domination and distance k-covering. We obtain results for maximal outerplanar graphs when k=2. A set S of vertices in a triangulation graph T is a distance 2-guarding set (or 2d-guarding set for short) if every face of T has a vertex adjacent to a vertex of S. We show that ⌊n/5⌋ (respectively, ⌊n/4⌋) vertices are sufficient to 2d-guard and 2d-dominate (respectively, 2d-cover) any n-vertex maximal outerplanar graph. We also show that these bounds are tight
Hyperopic Cops and Robbers
We introduce a new variant of the game of Cops and Robbers played on graphs,
where the robber is invisible unless outside the neighbor set of a cop. The
hyperopic cop number is the corresponding analogue of the cop number, and we
investigate bounds and other properties of this parameter. We characterize the
cop-win graphs for this variant, along with graphs with the largest possible
hyperopic cop number. We analyze the cases of graphs with diameter 2 or at
least 3, focusing on when the hyperopic cop number is at most one greater than
the cop number. We show that for planar graphs, as with the usual cop number,
the hyperopic cop number is at most 3. The hyperopic cop number is considered
for countable graphs, and it is shown that for connected chains of graphs, the
hyperopic cop density can be any real number in $[0,1/2].
Unified almost linear kernels for generalized covering and packing problems on nowhere dense classes
Let be a family of graphs, and let be nonnegative
integers. The \textsc{-Covering} problem asks whether for a
graph and an integer , there exists a set of at most vertices in
such that has no induced subgraph isomorphic to a
graph in , where is the -th power of . The
\textsc{-Packing} problem asks whether for a graph and
an integer , has induced subgraphs such that each
is isomorphic to a graph in , and for distinct , the distance between and in is larger than
.
We show that for every fixed nonnegative integers and every fixed
nonempty finite family of connected graphs, the
\textsc{-Covering} problem with and the
\textsc{-Packing} problem with
admit almost linear kernels on every nowhere dense class of graphs, and admit
linear kernels on every class of graphs with bounded expansion, parameterized
by the solution size . We obtain the same kernels for their annotated
variants. As corollaries, we prove that \textsc{Distance- Vertex Cover},
\textsc{Distance- Matching}, \textsc{-Free Vertex Deletion},
and \textsc{Induced--Packing} for any fixed finite family
of connected graphs admit almost linear kernels on every nowhere
dense class of graphs and linear kernels on every class of graphs with bounded
expansion. Our results extend the results for \textsc{Distance- Dominating
Set} by Drange et al. (STACS 2016) and Eickmeyer et al. (ICALP 2017), and the
result for \textsc{Distance- Independent Set} by Pilipczuk and Siebertz (EJC
2021).Comment: 38 page