An extended planar algorithm for maximum integral two-flow

On weighted efficient total domination

In a generalized Turán problem, we are given graphs $H$ and $F$ and seek to maximize the number of copies of $H$ in an $n$-vertex graph not containing $F$ as a subgraph. We consider generalized Turán problems where the host graph is planar. In particular, we obtain the order of magnitude of the maximum number of copies of a fixed tree in a planar graph containing no even cycle of length at most $2\ell$, for all $\ell$, $\ell \geqslant 1$. We also determine the order of magnitude of the maximum number of cycles of a given length in a planar $C_4$-free graph. An exact result is given for the maximum number of $5$-cycles in a $C_4$-free planar graph. Multiple conjectures are also introduced.

In a graph G a matching is a set of edges in which no two edges have a common endpoint. An induced matching is a matching in which no two edges are linked by an edge of G. The maximum induced matching (abbreviated MIM) problem is to find the maximum size of an induced matching for a given graph G. This problem is known to be NP-hard even on bipartite graphs or on planar graphs. We present a polynomial time algorithm which given a graph G either finds a maximum induced matching in G, or claims that the size of a maximum induced matching in G is strictly less than the size of a maximum matching in G. We show that the MIM problem is NP-hard on line-graphs, claw-free graphs, chair-free graphs, Hamiltonian graphs and r-regular graphs for r \geq 5. On the other hand, we present polynomial time algorithms for the MIM problem on (P5,Dm)-free graphs, on (bull, chair)-free graphs and on line-graphs of Hamiltonian graphs.

We show that for an integer k ≥ 2 and an n-vertex graph G without a K 3,3 (resp., K 5) minor, we can compute k induced subgraphs of G with treewidth ≤ 3k−4 (resp., ≤ 6k−7) in O(kn) (resp., O(kn+n 2)) time such that each vertex of G appears in exactly k − 1 of these subgraphs. This leads to practical polynomial-time approximation schemes for various maximum induced-subgraph problems on graphs without a K 3,3 or K 5 minor. The result extends a well-known result of Baker that there are practical polynomial-time approximation schemes for various maximum induced-subgraph problems on planar graphs.

A clique-coloring of a given graph G is a coloring of the vertices of G such that no maximal clique of size at least two is monocolored. The clique-chromatic number of G is the least number of colors for which G admits a clique-coloring. It has been proved that every planar graph is 3-clique colorable and every claw-free planar graph, different from an odd cycle, is 2-clique colorable. In this paper, we generalize these results to $$K_{3,3}$$ -minor free ( $$K_{3,3}$$ -subdivision free) graphs.

In the exact matching problem we are given a graph G, some of whose edges are colored red, and a positive integer k. The goal is to determine if G has a perfect matching, exactly k edges of which are red. More generally if the matching number of G is m=m(G), the goal is to find a matching with m edges, exactly k edges of which are red, or determine that no such matching exists. This problem is one of the few remaining problems that have efficient randomized algorithms (in fact, this problem is in RNC), but for which no polynomial time deterministic algorithm is known. Our first result shows that, in a sense, this problem is as close to being in P as one can get. We give a polynomial time deterministic algorithm that either correctly decides that no maximum matching has exactly k red edges, or exhibits a matching with m(G)−1 edges having exactly k red edges. Hence, the additive error is one. We also present an efficient algorithm for the exact matching problem in families of graphs for which this problem is known to be tractable. We show how to count the number of exact perfect matchings in K 3,3-minor free graphs (these include all planar graphs as well as many others) in O(n 3.19) worst case time. Our algorithm can also count the number of perfect matchings in K 3,3-minor free graphs in O(n 2.19) time.

Finding dominating induced matchings in [formula omitted]-free graphs in polynomial time

We study the dominating set problem in an online setting. An algorithm is required to guarantee competitiveness against an adversary that reveals the input graph one node at a time. When a node is revealed, the algorithm learns about the entire neighborhood of the node (including those nodes that have not yet been revealed). Furthermore, the adversary is required to keep the revealed portion of the graph connected at all times. We present an algorithm that achieves 2-competitiveness on trees. We also present algorithms that achieve 2.5-competitiveness on cactus graphs, (t-1)-competitiveness on K_{1,t}-free graphs, and Θ(√{Δ}) for maximum degree Δ graphs. We show that all of those competitive ratios are tight. Then, we study several more general classes of graphs, such as threshold, bipartite planar, and series-parallel graphs, and show that they do not admit competitive algorithms (i.e., when competitive ratio is independent of the input size). Previously, the dominating set problem was considered in a different input model (often together with the restriction of the input graph being always connected), where a vertex is revealed alongside its restricted neighborhood: those neighbors that are among already revealed vertices. Thus, conceptually, our results quantify the value of knowing the entire neighborhood at the time a vertex is revealed as compared to the restricted neighborhood. For instance, it was known in the restricted neighborhood model that 3-competitiveness is optimal for trees, whereas knowing the neighbors allows us to improve it to 2-competitiveness.

The total chromatic number of a graph $G$, denoted by $\chi′′(G)$, is the minimum number of colors needed to color the vertices and edges of $G$ such that no two adjacent or incident elements get the same color. It is known that if a planar graph $G$ has maximum degree $\Delta ≥ 9$, then $\chi′′(G) = \Delta + 1$. The join $K_1 \vee P_n$ of $K_1$ and $P_n$ is called a fan graph $F_n$. In this paper, we prove that if $G$ is a $F_5$-free planar graph with maximum degree 8, then $\chi′′(G) = 9$.

We develop a bipartite rigidity theory for bipartite graphs parallel to the classical rigidity theory for general graphs, and define for two positive integers k , l k,l the notions of ( k , l ) (k,l) -rigid and ( k , l ) (k,l) -stress free bipartite graphs. This theory coincides with the study of Babson–Novik’s balanced shifting restricted to graphs. We establish bipartite analogs of the cone, contraction, deletion, and gluing lemmas, and apply these results to derive a bipartite analog of the rigidity criterion for planar graphs. Our result asserts that for a planar bipartite graph G G its balanced shifting, G b G^b , does not contain K 3 , 3 K_{3,3} ; equivalently, planar bipartite graphs are generically ( 2 , 2 ) (2,2) -stress free. We also discuss potential applications of this theory to Jockusch’s cubical lower bound conjecture and to upper bound conjectures for embedded simplicial complexes.

A plane graph G is coupled k‐choosable if, for any list assignment L satisfying $|{{L}}({{x}})|= {{k}}$ for every ${{x}}\in {{V}}({{G}})\cup {{F}}({{G}})$, there is a coloring that assigns to each vertex and each face a color from its list such that any two adjacent or incident elements receive distinct colors. We prove that every plane graph is coupled 7‐choosable. We further show that maximal plane graphs, ${{K}}_{{4}}$‐minor free graphs, and plane graphs with maximum degree at most three are coupled 6‐choosable. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 27–44, 2008

Graph isomorphism is an important and widely studied computational problem with a yet unsettled complexity. However, the exact complexity is known for isomorphism of various classes of graphs. Recently, \cite{DLNTW09} proved that planar isomorphism is complete for log-space. We extend this result %of \cite{DLNTW09} further to the classes of graphs which exclude $K_{3,3}$ or $K_5$ as a minor, and give a log-space algorithm. Our algorithm decomposes $K_{3,3}$ minor-free graphs into biconnected and those further into triconnected components, which are known to be either planar or $K_5$ components \cite{Vaz89}. This gives a triconnected component tree similar to that for planar graphs. An extension of the log-space algorithm of \cite{DLNTW09} can then be used to decide the isomorphism problem. For $K_5$ minor-free graphs, we consider $3$-connected components. These are either planar or isomorphic to the four-rung mobius ladder on $8$ vertices or, with a further decomposition, one obtains planar $4$-connected components \cite{Khu88}. We give an algorithm to get a unique decomposition of $K_5$ minor-free graphs into bi-, tri- and $4$-connected components, and construct trees, accordingly. Since the algorithm of \cite{DLNTW09} does not deal with four-connected component trees, it needs to be modified in a quite non-trivial way.

We show that the reachability problem for directed graphs that are either K 3,3-free or K 5-free is in unambiguous log-space, UL ∩ coUL. This significantly extends the result of Bourke, Tewari, and Vinodchandran that the reachability problem for directed planar graphs is in UL ∩ coUL.

We study the problem of determining whether a given graph \(G=(V,E)\) admits a matching M whose removal destroys all odd cycles of G (or equivalently whether \(G-M\) is bipartite). This problem is equivalent to determine whether G admits a (2, 1)-coloring, which is a 2-coloring of V(G) in which each color class induces a graph of maximum degree at most 1. We determine a dichotomy related to the NP-completeness of such a decision problem, where it is NP-complete even for 3-colorable planar graphs of maximum degree 4, while it is linear-time solvable for graphs of maximum degree at most 3. In addition, we present polynomial-time algorithms for many graph classes including those in which every odd-cycle subgraph is a triangle, graphs having bounded dominating sets, and \(P_5\)-free graphs. Additionally, we show that this problem is fixed-parameter tractable when parameterized by the clique-width, which implies that it is polynomial-time solvable for many interesting graph classes, such as distance-hereditary, outerplanar, and chordal graphs.

This paper revisits design and analysis techniques for fixed parameter algorithms for Planar Dominating Set and other problems on planar structures. As our main result, we use new geometric arguments concerning treewidth-based algorithms to show that determining whether a planar graph G has a dominating set of size k can be solved in \(O(2^{16.4715 \sqrt{k}}+ n^3)\) steps. This result improves on the best known treewidth-based algorithm by Kanj and Perkovič that runs in time \(O(2^{27\sqrt{k}}n)\). Our main result nearly matches the new branchwidth-based algorithm for Planar Dominating Set by Fomin and Thilikos that runs in time \(O(2^{15.13 \sqrt{k}}k +n^3)\). Algorithms for other problems on planar structures are explored. In particular, we show that Planar Red/Blue Dominating Set can be solved in time \(O(2^{24.551 \sqrt{k}}n)\). This leads to the main results, namely, that faster parameterized algorithms can be obtained for a variety of problems that can be described by planar boolean formulae. This gives the best-known parameterized algorithms for Planar Vertex Cover, Planar Edge Dominating Set, and Face Cover.