A $k$-dimensional box is the cartesian product $R_1 \times R_2 \times ... \times R_k$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as $box(G)$, is the minimum integer $k$ such that $G$ can be represented as the intersection graph of a collection of $k$-dimensional boxes: that is two vertices are adjacent if and only if their corresponding boxes intersect. A circular arc graph is a graph that can be represented as the intersection graph of arcs on a circle.
Let $G$ be a circular arc graph with maximum degree $Δ$. We show that if $Δ<\lfloor \frac{n(α-1)}{2α}\rfloor$, $α\in \mathbb{N}$, $α\geq 2$ then $box(G) \leq α$. We also demonstrate a graph with boxicity $> α$ but with $Δ=n\frac{(α-1)}{2α}+\frac{n}{2α(α+1)}+(α+2)$. So the result cannot be improved substantially when $α$ is large. Let $r_{inf}$ be minimum number of arcs passing through any point on the circle with respect to some circular arc representation of $G$. We also show that for any circular arc graph $G$, $box(G) \leq r_{inf} + 1$ and this bound is tight. Given a family of arcs $F$ on the circle, the circular cover number $L(F)$ is the cardinality of the smallest subset $F'$ of $F$ such that the arcs in $F'$ can cover the circle. Maximum circular cover number $L_{max}(G)$ is defined as the maximum value of $L(F)$ obtained over all possible family of arcs $F$ that can represent $G$. We will show that if $G$ is a circular arc graph with $L_{max}(G)> 4$ then $box(G) \leq 3$.
In this short note, we relate the boxicity of graphs (and the dimension of posets) with their generalized coloring parameters. In particular, together with known estimates, our results imply that any graph with no $K_t$-minor can be represented as the intersection of $O(t^2\log t)$ interval graphs (improving the previous bound of $O(t^4)$), and as the intersection of $\tfrac{15}2 t^2$ circular-arc graphs.
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…
In this paper, we give a characterization of the class of all circular-arc graphs whose schemes are association. Moreover, all association schemes which are the scheme of a circular-arc graph are characterized, specially it is proved that they are Schurian.
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, chord…
We consider circular version of the famous Nelson-Hadwiger problem. It is know that 4 colors are necessary and 7 colors suffice to color the euclidean plane in such a way that points at distance one get different colors. In $r$-circular coloring we assign arcs of length one of a circle with a perimeter $r$ in such a way that points at distance one get disjoint arcs. In this paper we show the existence of $r$-circular coloring for $r=4+\frac{4\sqrt{3}}{3}\approx 6.30$. It is the first result with $r$-circular coloring of the plane with $r$ smaller than 7. We also show $r$-circular coloring of t…
In the system of approval voting, individuals vote for all candidates they find acceptable. Many approval voting situations can be modeled geometrically, and thus geometric concepts such as the piercing number have a natural interpretation. In this paper, we explore piercing numbers in the setting where voter preferences can be modeled by congruent arcs on a circle -- i.e., in fixed-length circular societies. Given a number of voters and the length of the voter preference arcs, we give bounds on the possible piercing number of the society. Further, we explore which piercing numbers are more li…
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 circular-arc graph is the intersection graph of arcs of a circle. It is a well-studied graph model with numerous natural applications. A certifying algorithm is an algorithm that outputs a certificate, along with its answer (be it positive or negative), where the certificate can be used to easily justify the given answer. While the recognition of circular-arc graphs has been known to be polynomial since the 1980s, no polynomial-time certifying recognition algorithm is known to date, despite such algorithms being found for many subclasses of circular-arc graphs. This is largely due to the fac…
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 isomorphism problem is known to be efficiently solvable for interval graphs, while for the larger class of circular-arc graphs its complexity status stays open. We consider the intermediate class of intersection graphs for families of circular arcs that satisfy the Helly property. We solve the isomorphism problem for this class in logarithmic space. If an input graph has a Helly circular-arc model, our algorithm constructs it canonically, which means that the models constructed for isomorphic graphs are equal.
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…
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…
We show that the nerve complex of n arcs in the circle is homotopy equivalent to either a point, an odd-dimensional sphere, or a wedge sum of spheres of the same even dimension. Moreover this homotopy type can be computed in time O(n log n). For the particular case of the nerve complex of evenly-spaced arcs of the same length, we determine the dihedral group action on homology, and we relate the complex to a cyclic polytope with n vertices. We give three applications of our knowledge of the homotopy types of nerve complexes of circular arcs. First, we use the connection to cyclic polytopes to …
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 …
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…
