146 research outputs found
Polygon-circle and word-representable graphs
We describe work on the relationship between the independently-studied polygon-circle graphs and word-representable graphs. A graph G = (V, E) is word-representable if there exists a word w over the alpha-bet V such that letters x and y form a subword of the form xyxy ⋯ or yxyx ⋯ iff xy is an edge in E. Word-representable graphs generalise several well-known and well-studied classes of graphs [S. Kitaev, A Comprehensive Introduction to the Theory of Word-Representable Graphs, Lecture Notes in Computer Science 10396 (2017) 36–67; S. Kitaev, V. Lozin, “Words and Graphs” Springer, 2015]. It is known that any word-representable graph is k-word-representable, that is, can be represented by a word having exactly k copies of each letter for some k dependent on the graph. Recognising whether a graph is word-representable is NP-complete ([S. Kitaev, V. Lozin, “Words and Graphs” Springer, 2015, Theorem 4.2.15]). A polygon-circle graph (also known as a spider graph) is the intersection graph of a set of polygons inscribed in a circle [M. Koebe, On a new class of intersection graphs, Ann. Discrete Math. (1992) 141–143]. That is, two vertices of a graph are adjacent if their respective polygons have a non-empty intersection, and the set of polygons that correspond to vertices in this way are said to represent the graph. Recognising whether an input graph is a polygon-circle graph is NP-complete [M. Pergel, Recognition of polygon-circle graphs and graphs of interval filaments is NP-complete, Graph-Theoretic Concepts in Computer Science: 33rd Int. Workshop, Lecture Notes in Computer Science, 4769 (2007) 238–247]. We show that neither of these two classes is included in the other one by showing that the word-representable Petersen graph and crown graphs are not polygon-circle, while the non-word-representable wheel graph W 5 is polygon-circle. We also provide a more refined result showing that for any k ≥ 3, there are k-word-representable graphs which are neither (k −1)-word-representable nor polygon-circle
Equivalence of the filament and overlap graphs of subtrees of limited trees
The overlap graphs of subtrees of a tree are equivalent to subtree filament
graphs, the overlap graphs of subtrees of a star are cocomparability graphs,
and the overlap graphs of subtrees of a caterpillar are interval filament
graphs. In this paper, we show the equivalence of many more classes of subtree
overlap and subtree filament graphs, and equate them to classes of complements
of cochordal-mixed graphs. Our results generalize the previously known results
mentioned above
Unit Grid Intersection Graphs: Recognition and Properties
It has been known since 1991 that the problem of recognizing grid
intersection graphs is NP-complete. Here we use a modified argument of the
above result to show that even if we restrict to the class of unit grid
intersection graphs (UGIGs), the recognition remains hard, as well as for all
graph classes contained inbetween. The result holds even when considering only
graphs with arbitrarily large girth. Furthermore, we ask the question of
representing UGIGs on grids of minimal size. We show that the UGIGs that can be
represented in a square of side length 1+epsilon, for a positive epsilon no
greater than 1, are exactly the orthogonal ray graphs, and that there exist
families of trees that need an arbitrarily large grid
Recognising the overlap graphs of subtrees of restricted trees is hard
The overlap graphs of subtrees in a tree (SOGs) generalise many other graphs classes with set representation characterisations. The complexity of recognising SOGs is open. The complexities of recognising many subclasses of SOGs are known. Weconsider several subclasses of SOGs by restricting the underlying tree. For a fixed integer , we consider:\begin{my_itemize} \item The overlap graphs of subtrees in a tree where that tree has leaves
\item The overlap graphs of subtrees in trees that can be derived from a given input tree by subdivision and have at least 3 leaves \item The overlap and intersection graphs of paths in a tree where that tree has maximum degree \end{my_itemize}
We show that the recognition problems of these classes are NP-complete. For all other parameters we get circle graphs, well known to be polynomially recognizable
Recognising the overlap graphs of subtrees of restricted trees is hard
The overlap graphs of subtrees in a tree (SOGs) generalise many other graphs classes with set representation characterisations. The complexity of recognising SOGs is open. The complexities of recognising many subclasses of SOGs are known. Weconsider several subclasses of SOGs by restricting the underlying tree. For a fixed integer , we consider:\begin{my_itemize} \item The overlap graphs of subtrees in a tree where that tree has leaves
\item The overlap graphs of subtrees in trees that can be derived from a given input tree by subdivision and have at least 3 leaves \item The overlap and intersection graphs of paths in a tree where that tree has maximum degree \end{my_itemize}
We show that the recognition problems of these classes are NP-complete. For all other parameters we get circle graphs, well known to be polynomially recognizable
Large induced subgraphs via triangulations and CMSO
We obtain an algorithmic meta-theorem for the following optimization problem.
Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an
integer. For a given graph G, the task is to maximize |X| subject to the
following: there is a set of vertices F of G, containing X, such that the
subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X)
models \phi.
Some special cases of this optimization problem are the following generic
examples. Each of these cases contains various problems as a special subcase:
1) "Maximum induced subgraph with at most l copies of cycles of length 0
modulo m", where for fixed nonnegative integers m and l, the task is to find a
maximum induced subgraph of a given graph with at most l vertex-disjoint cycles
of length 0 modulo m.
2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\
containing a planar graph, the task is to find a maximum induced subgraph of a
given graph containing no graph from \Gamma\ as a minor.
3) "Independent \Pi-packing", where for a fixed finite set of connected
graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G
with the maximum number of connected components, such that each connected
component of G[F] is isomorphic to some graph from \Pi.
We give an algorithm solving the optimization problem on an n-vertex graph G
in time O(#pmc n^{t+4} f(t,\phi)), where #pmc is the number of all potential
maximal cliques in G and f is a function depending of t and \phi\ only. We also
show how a similar running time can be obtained for the weighted version of the
problem. Pipelined with known bounds on the number of potential maximal
cliques, we deduce that our optimization problem can be solved in time
O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with
polynomial number of minimal separators
On the helly property of some intersection graphs
An EPG graph G is an edge-intersection graph of paths on a grid. In this
doctoral thesis we will mainly explore the EPG graphs, in particular B1-EPG graphs.
However, other classes of intersection graphs will be studied such as VPG, EPT and
VPT graph classes, in addition to the parameters Helly number and strong Helly
number to EPG and VPG graphs. We will present the proof of NP-completeness
to Helly-B1-EPG graph recognition problem. We investigate the parameters Helly
number and the strong Helly number in both graph classes, EPG and VPG in order
to determine lower bounds and upper bounds for this parameters. We completely
solve the problem of determining the Helly and strong Helly numbers, for Bk-EPG,
and Bk-VPG graphs, for each value k.
Next, we present the result that every Chordal B1-EPG graph is simultaneously
in the VPT and EPT graph classes. In particular, we describe structures that occur
in B1-EPG graphs that do not support a Helly-B1-EPG representation and thus we
define some sets of subgraphs that delimit Helly subfamilies. In addition, features
of some non-trivial graph families that are properly contained in Helly-B1 EPG are
also presented.EPG é um grafo de aresta-interseção de caminhos sobre uma grade.
Nesta tese de doutorado exploraremos principalmente os grafos EPG, em particular
os grafos B1-EPG. Entretanto, outras classes de grafos de interseção serão estu dadas, como as classes de grafos VPG, EPT e VPT, além dos parâmetros número
de Helly e número de Helly forte nos grafos EPG e VPG. Apresentaremos uma
prova de NP-completude para o problema de reconhecimento de grafos B1-EPG Helly. Investigamos os parâmetros número de Helly e o número de Helly forte nessas
duas classes de grafos, EPG e VPG, a fim de determinar limites inferiores e superi ores para esses parâmetros. Resolvemos completamente o problema de determinar o
número de Helly e o número de Helly forte para os grafos Bk-EPG e Bk-VPG, para
cada valor k.
Em seguida, apresentamos o resultado de que todo grafo B1-EPG Chordal está
simultaneamente nas classes de grafos VPT e EPT. Em particular, descrevemos
estruturas que ocorrem em grafos B1-EPG que não suportam uma representação
B1-EPG-Helly e assim definimos alguns conjuntos de subgrafos que delimitam sub famílias Helly. Além disso, também são apresentadas características de algumas
famílias de grafos não triviais que estão propriamente contidas em B1-EPG-Hell
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