11 research outputs found
Planar projections of graphs
We introduce and study a new graph representation where vertices are embedded
in three or more dimensions, and in which the edges are drawn on the
projections onto the axis-parallel planes. We show that the complete graph on
vertices has a representation in planes. In 3
dimensions, we show that there exist graphs with edges that can be
projected onto two orthogonal planes, and that this is best possible. Finally,
we obtain bounds in terms of parameters such as geometric thickness and linear
arboricity. Using such a bound, we show that every graph of maximum degree 5
has a plane-projectable representation in 3 dimensions.Comment: Accepted at CALDAM 202
Quotient-4 Cordial Labeling Of Some Caterpillar And Lobster Graphs
Let G (V, E) be a simple graph of order p and size q. Let φ: V (G) Z5 – {0} be a function. For each edge set E (G) define the labeling *:E (G)Z4 by *(uv)= (mod 4) where (u)(v). The function is called Quotient-4 cordial labeling of G if |vφ(i) – vφ(j)| ≤ 1, , j, ij where vφ(x) denote the number of vertices labeled with x and |eφ(k) – eφ(l)| ≤ 1, ,,, where eφ(y) denote the number of edges labeled with y. Here some caterpillar graphs such as star graph (Sn), Bistar graph (Bn,n), Pn [N] graph, Pn [No] graph, Pn [Ne] graph, Twig graph (Tm), (Pn K1, r), S(Sn), S(Bn,n), S(Pn [N]), S(Pn [No]), S(Pn [Ne]), S(Tm) and S(Pn K1, r) graph proved to be quotient-4 cordial graphs
Three ways to cover a graph
We consider the problem of covering an input graph with graphs from a
fixed covering class . The classical covering number of with respect to
is the minimum number of graphs from needed to cover the edges of
without covering non-edges of . We introduce a unifying notion of three
covering parameters with respect to , two of which are novel concepts only
considered in special cases before: the local and the folded covering number.
Each parameter measures "how far'' is from in a different way. Whereas
the folded covering number has been investigated thoroughly for some covering
classes, e.g., interval graphs and planar graphs, the local covering number has
received little attention.
We provide new bounds on each covering number with respect to the following
covering classes: linear forests, star forests, caterpillar forests, and
interval graphs. The classical graph parameters that result this way are
interval number, track number, linear arboricity, star arboricity, and
caterpillar arboricity. As input graphs we consider graphs of bounded
degeneracy, bounded degree, bounded tree-width or bounded simple tree-width, as
well as outerplanar, planar bipartite, and planar graphs. For several pairs of
an input class and a covering class we determine exactly the maximum ordinary,
local, and folded covering number of an input graph with respect to that
covering class.Comment: 20 pages, 4 figure
Beyond circular-arc graphs -- recognizing lollipop graphs and medusa graphs
In 1992 Bir\'{o}, Hujter and Tuza introduced, for every fixed connected graph
, the class of -graphs, defined as the intersection graphs of connected
subgraphs of some subdivision of . Recently, quite a lot of research has
been devoted to understanding the tractability border for various computational
problems, such as recognition or isomorphism testing, in classes of -graphs
for different graphs . In this work we undertake this research topic,
focusing on the recognition problem. Chaplick, T\"{o}pfer, Voborn\'{\i}k, and
Zeman showed, for every fixed tree , a polynomial-time algorithm recognizing
-graphs. Tucker showed a polynomial time algorithm recognizing -graphs
(circular-arc graphs). On the other hand, Chaplick at al. showed that
recognition of -graphs is -hard if contains two different cycles
sharing an edge.
The main two results of this work narrow the gap between the -hard and
cases of -graphs recognition. First, we show that recognition of
-graphs is -hard when contains two different cycles. On the other
hand, we show a polynomial-time algorithm recognizing -graphs, where is
a graph containing a cycle and an edge attached to it (-graphs are called
lollipop graphs). Our work leaves open the recognition problems of -graphs
for every unicyclic graph different from a cycle and a lollipop. Other
results of this work, which shed some light on the cases that remain open, are
as follows. Firstly, the recognition of -graphs, where is a fixed
unicyclic graph, admits a polynomial time algorithm if we restrict the input to
graphs containing particular holes (hence recognition of -graphs is probably
most difficult for chordal graphs). Secondly, the recognition of medusa graphs,
which are defined as the union of -graphs, where runs over all unicyclic
graphs, is -complete
-free Partition and Cover Numbers and Application
-free graphs-- also known as cographs, complement-reducible graphs, or hereditary Dacey graphs--have been well studied in graph theory.
Motivated by computer science and information theory applications, our work encodes (flat) joint probability distributions and Boolean functions as bipartite graphs and studies bipartite -free graphs.
For these applications, the graph properties of edge partitioning and covering a bipartite graph using the minimum number of these graphs are particularly relevant.
Previously, such graph properties have appeared in leakage-resilient cryptography and (variants of) coloring problems.
Interestingly, our covering problem is closely related to the well-studied problem of product/Prague dimension of loopless undirected graphs, which allows us to employ algebraic lower-bounding techniques for the product/Prague dimension.
We prove that computing these numbers is \npol-complete, even for bipartite graphs.
We establish a connection to the (unsolved) Zarankiewicz problem to show that there are bipartite graphs with size- partite sets such that these numbers are at least , for .
Finally, we accurately estimate these numbers for bipartite graphs encoding well-studied Boolean functions from circuit complexity, such as set intersection, set disjointness, and inequality.
For applications in information theory and communication \& cryptographic complexity, we consider a system where a setup samples from a (flat) joint distribution and gives the participants, Alice and Bob, their portion from this joint sample.
Alice and Bob\u27s objective is to non-interactively establish a shared key and extract the left-over entropy from their portion of the samples as independent private randomness.
A genie, who observes the joint sample, provides appropriate assistance to help Alice and Bob with their objective.
Lower bounds to the minimum size of the genie\u27s assistance translate into communication and cryptographic lower bounds.
We show that (the of) the -free partition number of a graph encoding the joint distribution that the setup uses is equivalent to the size of the genie\u27s assistance.
Consequently, the joint distributions corresponding to the bipartite graphs constructed above with high -free partition numbers correspond to joint distributions requiring more assistance from the genie.
As a representative application in non-deterministic communication complexity, we study the communication complexity of nondeterministic protocols augmented by access to the equality oracle at the output.
We show that (the of) the -free cover number of the bipartite graph encoding a Boolean function is equivalent to the minimum size of the nondeterministic input required by the parties (referred to as the communication complexity of in this model).
Consequently, the functions corresponding to the bipartite graphs with high -free cover numbers have high communication complexity.
Furthermore, there are functions with communication complexity close to the \naive protocol where the nondeterministic input reveals a party\u27s input.
Finally, the access to the equality oracle reduces the communication complexity of computing set disjointness by a constant factor in contrast to the model where parties do not have access to the equality oracle.
To compute the inequality function, we show an exponential reduction in the communication complexity, and this bound is optimal.
On the other hand, access to the equality oracle is (nearly) useless for computing set intersection
Non-crossing shortest paths in planar graphs with applications to max flow, and path graphs
This thesis is concerned with non-crossing shortest paths in planar graphs with applications to st-max flow vitality and path graphs.
In the first part we deal with non-crossing shortest paths in a plane graph G, i.e., a planar graph with a fixed planar embedding, whose extremal vertices lie on the same face of G. The first two results are the computation of the lengths of the non-crossing shortest paths knowing their union, and the computation of the union in the unweighted case. Both results require linear time and we use them to describe an efficient algorithm able to give an additive guaranteed approximation of edge and vertex vitalities with respect to the st-max flow in undirected planar graphs, that is the max flow decrease when the edge/vertex is removed from the graph. Indeed, it is well-known that the st-max flow in an undirected planar graph can be reduced to a problem of non-crossing shortest paths in the dual graph. We conclude this part by showing that the union of non-crossing shortest paths in a plane graph can be covered with four forests so that each path is contained in at least one forest.
In the second part of the thesis we deal with path graphs and directed path graphs, where a (directed) path graph is the intersection graph of paths in a (directed) tree. We introduce a new characterization of path graphs that simplifies the existing ones in the literature. This characterization leads to a new list of local forbidden subgraphs of path graphs and to a new algorithm able to recognize path graphs and directed path graphs. This algorithm is more intuitive than the existing ones and does not require sophisticated data structures