    Map graphs having witnesses of large girth

A half-square of a bipartite graph B=(X,Y,E_B) has one color class of B as vertex set, say X; two vertices are adjacent whenever they have a common neighbor in Y. If G=(V,E_G) is the half-square of a planar bipartite graph B=(V,W,E_B), then G is called a map graph, and B is a witness of G. Map graphs generalize planar graphs, and have been introduced and investigated by Chen, Grigni and Papadimitriou [STOC 1998, J. ACM 2002]. They proved that recognizing map graphs is in NP by proving the existence of a witness. Soon later, Thorup [FOCS 1998] claimed that recognizing map graphs is in P, by providing an Ω(n^120)-time algorithm for n-vertex input graphs. In this note, we give good characterizations and efficient recognition for half-squares of bipartite graphs with girth at least a given integer g> 8. It turns out that map graphs having witnesses of girth at least g are precisely the graphs whose vertex-clique incidence bipartite graph is planar and of girth at least g. Our structural characterization implies an O(n^2m)-time algorithm for recognizing if a given n-vertex m-edge graph G is such a map graph.

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1 Introduction and preliminaries

A map graph, introduced and investigated by Chen et al. ChenGP98 ; ChenGP02 , is the intersection graph of simply-connected and interior-disjoint regions of the Euclidean plane. More precisely, a map of a graph is a function taking each vertex to a closed disc homeomorph (the regions) in the plane, such that all , , are interior-disjoint, and two distinct vertices and of are adjacent if and only if the boundaries of and intersect. A map graph is one having a map. Map graphs are interesting as they generalize planar graphs in a very natural way. Some applications of map graphs have been addressed in ChenHK99 . Papers dealing with hard problems in map graphs include Chen ; DemaineFHT ; DemaineH ; EickmeyerK17 ; FominLS . Certain map graphs are related to 1-planar graphs Brandenburg15 ; Brandenburg18 ; ChenGP06 , a relevant topic in graph drawing.

In ChenGP98 ; ChenGP02 , the notion of half-squares of bipartite graphs also has been introduced in order to give a graph-theoretical characterization of map graphs. The square of a graph , denoted , is obtained from by adding new edges between two distinct vertices whenever their distance in is two. Then, is called a square root of . Given a bipartite graph , the subgraphs of the square induced by the color classes and , and , are called the two half-squares of . It turns out that map graphs are exactly half-squares of planar bipartite graphs ChenGP98 ; ChenGP02 . If is a map graph and is a planar bipartite graph such that , then is a witness of . It is shown in ChenGP98 ; ChenGP02 that an -vertex graph is a map graph if and only if it has a witness with , implying that recognizing map graphs is in . Soon later, Thorup Thorup98 claimed that recognizing map graphs is in

, by providing a polynomial-time algorithm. (Thorup did not give the running time explicitly, but it is estimated to be roughly

with being the vertex number of the input graph; cf. ChenGP02 .) Thorup’s algorithm is very complex and highly non-combinatorial. Given the very high polynomial degree in Thorup’s running time, the most discussed problem concerning map graphs is whether there is a faster recognition algorithm for map graphs. One direction in attacking this problem is to consider map graphs with restricted witness. Recently, in MnichRS16 , it is shown that map graphs with outerplanar witness and map graphs with tree witness can be recognized in linear time. (We remark that Thorup’s algorithm cannot be used to recognize map graphs having witnesses with additional properties.)

In this note we consider map graphs with girth constrained witness. Where, the girth of a graph is the length of a shortest cycle in the graph; the girth of a tree is . Our motivation is the observation that every planar graph is a map graph with a witness of girth at least six. Indeed, if is a planar graph, then the subdivision of , i.e., the vertex-edge incidence bipartite graph obtained from by replacing each edge by a path of length two, has girth at least six, and clearly . Note, however, that a clique of arbitrary size is a map graph with a star witness. So, in this context, the vertex-clique incidence bipartite graph is more useful than the subdivision.

Definition 1

Let be an arbitrary graph.

• The bipartite graph with is the subdivision, also the vertex-edge incidence bipartite graph, of .

• Let denote the set of all maximal cliques of . The bipartite graph with is the vertex-clique incidence bipartite graph of .

Note that subdivisions and vertex-clique incidence graphs of triangle-free graphs coincide. It is quite easy to see that, for every graph , and . Thus, in the context of map graphs, it is natural to ask for a given graph whether , respectively, , is planar. While it is well known that is planar if and only if is planar, the situation for is not clear yet.

Which (map) graphs have planar vertex-clique incidence bipartite graphs ?

The answer for the case when the vertex-clique incidence bipartite graph is a tree has been recently found by the following theorem.

Theorem 1 (LeL17 ; MnichRS16 )

A connected graph is a map graph with a tree witness if and only if it is a block graph, if and only if its vertex-clique incidence bipartite graph is a tree.

Where a block graph is one in which every maximal -connected subgraph (the blocks) is a clique. Equivalently, a block graph is a diamond-free chordal graph. (All terms used will given below.) As a consequence, map graphs with tree witness can be recognized in linear time.

In section 2 we will characterize half-squares of (not necessarily planar) bipartite graphs of large girth. Our structural results imply efficient polynomial time recognition for these half-squares. In section 3 we will consider map graphs having witnesses of large girth. It turns out that map graphs having witnesses of large girth (at least eight) admit a similar characterization as in case of tree witnesses stated in Theorem 1 above. As a consequence, we will see that such map graphs can be recognized in cubic time.

All graphs considered are simple and connected. The complete graph on vertices, the complete bipartite graph with vertices in one color class and vertices in the other color class, the cycle with vertices are denoted , and , respectively. A is also called a triangle, a complete bipartite graph is also called a star. The diamond, denoted , is the graph obtained from the by deleting an edge.

Let be a graph. -free graphs are those having no induced subgraphs isomorphic to . Chordal graphs are precisely the -free graphs, . It is well known that block graphs are precisely the diamond-free chordal graphs.

The neighborhood of a vertex in , denoted , is the set of all vertices in adjacent to ; if the context is clear, we simply write .

For a subset , is the subgraph of induced by , and stands for . We write for bipartite graphs with a bipartition into stable sets (color classes) and . If is a map graph and a witness of , then we call the set of vertices and the set of points.

2 Half-squares of bipartite graphs with girth constraints

Recall that every graph is a half-square of a bipartite graph with girth at least six. In this section, we will characterize those graphs that are half-squares of bipartite graphs with large girth. The following facts are easy to verify.

Observation 1

Let be a (not necessarily planar) bipartite graph and let . Then

• every induced cycle , , in stems from an induced cycle in . That is, if is an induced cycle in , then there are points in such that is an induced cycle in ;

• every triangle in stems from an induced or from a in .

In this section, we prove the following characterizations of half-squares of girth constrained bipartite graphs.

Theorem 2

Let be an integer. The following statements are equivalent for every graph .

• is half-square of a bipartite graph with girth at least ;

• is diamond-free and -free for every ;

• The vertex-clique incidence bipartite graph of has girth at least .

In particular,

• a graph is half-square of a bipartite graph with girth at least eight if and only if it is diamond-free, and

• a graph is half-square of a bipartite graph with girth at least ten if and only if it is diamond-free and -free.

It is also interesting to observe that, as grows larger, Theorem 2 gets closer and closer to Theorem 1 on map graphs with tree witness. (Recall that block graphs are diamond-free and -free for all , and a connected graph of girth is a tree.)

Proof. [of Theorem 2]

(i) (ii): Let for some bipartite graph of girth at least . Then, by Observation 1 (i), cannot contain any induced cycle for (otherwise, would contain an induced cycle of length ). Furthermore, by Observation 1 (ii), cannot contain an induced diamond. Otherwise, would contain an induced cycle of length (if one of the two triangles of the diamond stems from a in ) or an induced cycle of length (if both triangles of the diamond stem from stars in ).

(ii) (iii): Let have no induced diamond and no induced , . Recall that . For notational simplicity, write .

Claim 1.  is -free. If not, let be two points belonging to an induced in . Since and are two distinct maximal cliques in , there is a vertex not adjacent to a vertex in . But then , and two vertices in induce a diamond in .

Claim 2.  is -free. If not, let be an induced in . Then, in , is adjacent to two vertices of the maximal clique . Since , there exists a vertex in not adjacent to in . But then contains an induced diamond induced by and .

Now, consider a shortest cycle in , , of length . By Claims 1 and 2, . Moreover, by the minimality of , for (indices taken modulo ). Thus, induce an cycle of length in , hence . That is, has girth .

(iii) (i): This implication is obvious as for any graph , . ∎

Theorem 2 has algorithmic implications for recognizing half-squares of girth constrained bipartite graphs. In the remainder of this note, and denotes the vertex number and the edge number, respectively, of the graphs considered.

Half-squares of bipartite graphs of girth at least eight

These graphs are precisely the diamond-free graphs, and hence can be recognized in time EisenbrandG04 . Note that in a diamond-free graph, each edge belongs to exactly one maximal clique, hence there are at most maximal cliques. Since all maximal cliques in a graph can be listed in time per generated clique TsukiyamaIAS77 ; ChibaN85 , we can list all maximal cliques in in time . Thus, assuming is diamond-free, the vertex-clique incidence bipartite graph of can be constructed in time . Thus, we obtain

Corollary 1

Given an -vertex -edge graph , it can be recognized in time if is half-square of a bipartite graph with girth at least . If so, such a bipartite graph can be constructed in time.

Half-squares of bipartite graphs of girth at least 2t

Let be an integer. Half-squares of bipartite graphs of girth at least are precisely the graphs with is of girth at least . Note that such graphs are diamond-free and -free (Theorem 2 (ii)). Thus, we first recognize if the given graph is diamond-free and -free in time  EschenHSS11 . If so, we list all maximal cliques in , there are at most , in the same time complexity EschenHSS11 , to construct the vertex-clique incidence bipartite graph of . Since has vertices, the girth of (the bipartite graph) can be computed in time ItaiR78 ; YusterZ97 . Thus we obtain

Corollary 2

Given an -vertex graph and an integer , it can be recognized in time if is half-square of a bipartite graph with girth at least . If so, such a bipartite graph can be constructed in the same time complexity.

It is interesting to note that half-squares of bipartite graphs with girth at least ten, i.e., (diamond, )-free graphs, have been very recently discussed in the context of social networks: In FoxRSWW18 , -closed graphs are introduced as those graphs, in which every two vertices with at least common neighbors are adjacent. Thus, -closed graphs are precisely the (diamond, )-free graphs, i.e., the half-squares of bipartite graphs with girth at least ten.

3 Map graphs having witnesses with girth constraints

While any graph is the half-square of a bipartite graph of girth at least six, not every map graph has a witness of girth at least six. Such a map graph admitting no witness of girth at least six is depicted in Figure 1 below.

Notice that every planar graph is a map graph with a witness of girth at least six (e.g., its subdivision). So, map graphs admitting witnesses of girth at least six properly include all planar graphs. We are not able to characterize and recognize map graphs having witnesses of girth at least six.

In this section, we deal with map graphs having witnesses of girth at least eight. Note that not every planar graph has a witness of girth at least eight; the graph depicted in Figure 2 is diamond-free, hence it is the half-square of a bipartite graph of girth at least eight. However, as we will see (Theorem 3), this diamond-free planar graph does not have a witness of girth at least eight.

In discussing witnesses of girth at least eight, the following fact is useful; it has been observed and proved in LeL17 . To make the paper self-contained, we include the proof here.

Lemma 1

Let for some (not necessarily planar) bipartite graph . If has no induced cycle of length six, then every maximal clique in stems from a star in , i.e., there is a point such that .

Proof.  Suppose to the contrary that there is some clique in such that, for any point , . Choose a point where is maximal. Let . Since is a clique in , there is a point adjacent to and some vertices in . Choose such a point with is maximal. By the choice of , there is a vertex . Again, since is a clique in , there is a point adjacent to both and . By the choice of , there is a vertex non-adjacent to . But then and induce a in , a contradiction. Thus, there must be a point such that , and therefore by the maximality of , . ∎

In this section, we prove the following Theorem 3, characterizing map graphs having witnesses of large girth. Basically, Theorem 3 is Theorem 2 with the additional planarity condition on the vertex-clique incidence bipartite graph.

Theorem 3

Let be an integer. The following statements are equivalent for every graph .

• is a map graph having a witness of girth at least .

• is diamond-free and -free for every , and the vertex-clique incidence bipartite graph of is planar.

• The vertex-clique incidence bipartite graph of is planar and has girth at least .

Proof.

(i) (ii): Let be a planar bipartite graph of girth at least and with minimal number of points such that . First, it follows from Theorem 2, part (i) (ii), that is diamond-free and -free for every . Next, by Lemma 1, for every maximal clique in there is a point of such that . Since is minimal, such a point is unique: If there were another point with , then would be a planar bipartite graph of girth at least and still satisfy with fewer number of points than . Thus,

 ϕ:BG→B, with ϕ(v)=v, v∈V, and ϕ(Q)=wQ, Q∈C(G),

is an injective function. Moreover,

 vQ∈EBG⇔v∈Q=NB(wQ)⇔ϕ(v)ϕ(Q)=vwQ∈EBG,

that is, is isomorphic to an induced subgraph of . Hence is planar. (In fact, as , is indeed isomorphic to .)

(ii) (iii): This implication follows immediately from Theorem 2, part (ii) (iii).

(iii) (i): This implication is obvious as . ∎

Note that Theorem 3 is not true in case : The graph on the left side of Figure 2, as a planar graph, does admit a witness of girth six but its vertex-clique incidence bipartite graph is not planar; it contains as a minor.

Like Theorem 2, one may observe the ‘convergence behavior’ of Theorem 3 depending on ; it ‘converges’ to Theorem 1 as goes to infinity. We now derive efficient recognition algorithm from Theorem 3 for map graphs having witnesses of large girth. We use the fact that map graphs with vertices have at most many maximal cliques ChenGP02 .

Map graphs having witnesses of girth at least eight

Let be a map graph with a witness of girth at least . Since all maximal cliques in a graph can be listed in time per generated clique TsukiyamaIAS77 ; ChibaN85 , we can list all maximal cliques in in time to obtain the vertex-clique incidence bipartite graph . Since has vertices, checking planarity of can be done in time, and computing the girth of (planar) can be done in time LackiS11 . Thus, we obtain

Corollary 3

Map graphs having witnesses of girth at least can be recognized in time . If so, such a witness can be constructed with the same time complexity.

Map graphs having witnesses of girth at least 2t

Let be an integer. Map graphs having witnesses of girth at least are particularly diamond-free and -free. Thus, we first recognize if the given graph is diamond-free and -free in time  EschenHSS11 . If so, we list all maximal cliques in , there are at most , in the same time complexity EschenHSS11 , to construct the vertex-clique incidence bipartite graph of . Since has at most vertices, checking planarity of can be done in time, and computing the girth of (planar) can be done in time LackiS11 . Thus we obtain

Corollary 4

Let be an integer. Map graphs having witnesses of girth at least can be recognized in time . If so, such a witness can be constructed with the same time complexity.

4 Conclusion

In this note we consider half-squares of girth constrained bipartite graphs. Given an integer , we characterize and efficiently recognize half-squares of bipartite graphs with girth at least . We show that map graphs having witnesses with girth at least are exactly the graphs for which the vertex-clique incidence bipartite graph is planar and of girth at least . Hence map graphs having witnesses of girth at least can be recognized in time. The recognition and characterization problems of map graphs having witnesses of girth at least six remain unsolved.

• Which map graphs admit witnesses of girth at least ? Besides planar graphs, as pointed out by a referee, the so-called triangulated NIC-planar graphs, investigated in BachmaierBHNR17 , are examples of map graphs admitting witnesses of girth six.

• Is there an efficient combinatorial polynomial-time recognition algorithm for map graphs having witnesses of girth at least ?

Recall that the class of map graphs having witnesses of girth at least strictly lies between planar graphs and map graphs, and thus it is an interesting graph class.