Gromov hyperbolicity is an interesting geometric property, and so it is natural to study it in the context of geometric graphs. It measures the tree-likeness of a graph from a metric viewpoint. In particular, we are interested in circular-arc graphs, which is an important class of geometric intersection graphs. In this paper we give sharp bounds for the hyperbolicity constant of (finite and infinite) circular-arc graphs. Moreover, we obtain bounds for the hyperbolicity constant of the complement and line of any circular-arc graph. In order to do that, we obtain new results about regular, chordal and line graphs which are interesting by themselves.
Addressing a question of Zaremsky, we give conditions on a finite simplicial graph which guarantee that the associated matching arc complex is connected and hyperbolic.
Catch digraphs was introduced by Hiroshi Maehara in 1984 as an analog of intersection graphs where a family of pointed sets represents a digraph. After that Prisner continued his research particularly on interval catch digraphs by characterizing them diasteroidal triple free. It has numerous applications in the field of real world problems like network technology and telecommunication operations. Recently, we characterized three important subclasses of interval catch digraphs. In this article we introduce a new class of catch digraphs, namely circular-arc catch digraphs. The definition is same…
A Helly circular-arc graph is the intersection graph of a set of arcs on a circle having the Helly property. We introduce essential obstacles, which are a refinement of the notion of obstacles, and prove that essential obstacles are precisely the minimal forbidden induced circular-arc subgraphs for the class of Helly circular-arc graphs. We show that it is possible to find in linear time, in any given obstacle, some minimal forbidden induced subgraph for the class of Helly circular-arc graphs contained as an induced subgraph. Moreover, relying on an existing linear-time algorithm for finding i…
Circular-arc graphs are intersection graphs of arcs on the circle. The aim of our work is to present a polynomial time algorithm testing whether two circular-arc graphs are isomorphic. To accomplish our task we construct decomposition trees, which are the structures representing all normalized intersection models of circular-arc graphs. Normalized models reflect the neighbourhood relation in circular-arc graphs and can be seen as their canonical representations; in particular, every intersection model can be easily transformed into a normalized one. Our work adapts and appropriately extends th…
In 1969, Alan Tucker characterized proper circular-arc graphs as those graphs whose augmented adjacency matrices have the circularly compatible ones property. Moreover, he also found a polynomial-time algorithm for deciding whether any given augmented adjacency matrix has the circularly compatible ones property. These results allowed him to devise the first polynomial-time recognition algorithm for proper circular-arc graphs. However, as Tucker himself remarks, he did not solve the problems of finding a structure theorem and an efficient recognition algorithm for the circularly compatible ones…
A graph is said to be circular-arc if the vertices can be associated with arcs of a circle so that two vertices are adjacent if and only if the corresponding arcs overlap. It is proved that the isomorphism of circular-arc graphs can be tested by the Weisfeiler-Leman algorithm after individualization of two vertices.
In this short note, we show two NP-completeness results regarding the \emph{simultaneous representation problem}, introduced by Lubiw and Jampani. The simultaneous representation problem for a given class of intersection graphs asks if some $k$ graphs can be represented so that every vertex is represented by the same interval in each representation. We prove that it is NP-complete to decide this for the class of interval and circular-arc graphs in the case when $k$ is a part of the input and graphs are not in a sunflower position.
The partial representation extension problem generalizes the recognition problem for classes of graphs defined in terms of vertex representations. We exhibit circular-arc graphs as the first example of a graph class where the recognition is polynomially solvable while the representation extension problem is NP-complete. In this setting, several arcs are predrawn and we ask whether this partial representation can be completed. We complement this hardness argument with tractability results of the representation extension problem on various subclasses of circular-arc graphs, most notably on all v…
We study the dominating set reconfiguration problem with the token sliding rule. It consists, given a graph G=(V,E) and two dominating sets D_s and D_t of G, in determining if there exists a sequence S=<D_1:=D_s,...,D_l:=D_t> of dominating sets of G such that for any two consecutive dominating sets D_r and D_{r+1} with r<t, D_{r+1}=(D_r\ u) U v, where uv is an edge of G. In a recent paper, Bonamy et al studied this problem and raised the following questions: what is the complexity of this problem on circular arc graphs? On circle graphs? In this paper, we answer both questions by proving that …
Each hereditary property can be characterized by its set of minimal obstructions; these sets are often unknown, or known but infinite. By allowing extra structure it is sometimes possible to describe such properties by a finite set of forbidden objects. This has been studied most intensely when the extra structure is a linear ordering of the vertex set. For instance, it is known that a graph G is $k$-colourable if and only if $V(G)$ admits a linear ordering $\le$ with no vertices $v_1 \le \cdots \le v_{k+1}$ such that $v_i v_{i+1} \in E(G)$ for every $i \in \{ 1, \dots, k \}$. In this paper, w…
A graph is concave-round if its vertices can be circularly enumerated so that the closed neighbourhood of each vertex is an interval in the enumeration. In this work, we give a minimal forbidden induced subgraph characterization for the class of concave-round graphs, solving a problem posed by Bang-Jensen, Huang, and Yeo [SIAM J Discrete Math, 13:179--193, 2000]. In addition, we show that it is possible to find one such forbidden induced subgraph in linear time in any given graph that is not concave-round. As part of the analysis, we obtain characterizations by minimal forbidden submatrices fo…
