Algorithms for outerplanar graph roots and graph roots of pathwidth at most 2

Deciding whether a given graph has a square root is a classical problem that has been studied extensively both from graph theoretic and from algorithmic perspectives. The problem is NP-complete in general, and consequently substantial effort has been dedicated to deciding whether a given graph has a square root that belongs to a particular graph class. There are both polynomial-time solvable and NP-complete cases, depending on the graph class. We contribute with new results in this direction. Given an arbitrary input graph G, we give polynomial-time algorithms to decide whether G has an outerplanar square root, and whether G has a square root that is of pathwidth at most 2

A generalization of the Art Gallery Theorem

Several domination results have been obtained for maximal outerplanar graphs (mops). The classical domination problem is to minimize the size of a set $S$ of vertices of an $n$-vertex graph $G$ such that $G - N[S]$, the graph obtained by deleting the closed neighborhood of $S$, contains no vertices. A classical result of Chv\'{a}tal, the Art Gallery Theorem, tells us that the minimum size is at most $n/3$ if $G$ is a mop. Here we consider a modification by allowing $G - N[S]$ to have a maximum degree of at most $k$. Let $\iota_k(G)$ denote the size of a smallest set $S$ for which this is achieved. If $n \le 2k+3$, then trivially $\iota_k(G) \leq 1$. Let $G$ be a mop on $n \ge \max\{5,2k+3\}$ vertices, $n_2$ of which are of degree~$2$. Sharp bounds on $\iota_k(G)$ have been obtained for $k = 0$ and $k = 1$, namely $\iota_{0}(G) \le \min\{\frac{n}{4},\frac{n+n_2}{5},\frac{n-n_2}{3}\}$ and $\iota_1(G) \le \min\{\frac{n}{5},\frac{n+n_2}{6},\frac{n-n_2}{3}\}$. We prove that $\iota_{k}(G) \le \min\{\frac{n}{k+4},\frac{n+n_2}{k+5},\frac{n-n_2}{k+2}\}$ for any $k \ge 0$, and that this bound is sharp. We also prove that $\frac{n-n_2}{2}$ is a sharp upper bound on the domination number of $G$.Comment: arXiv admin note: substantial text overlap with arXiv:1903.1229

Isolation of squares in graphs

Given a set $\mathcal{F}$ of graphs, we call a copy of a graph in $\mathcal{F}$ an $\mathcal{F}$-graph. The $\mathcal{F}$-isolation number of a graph $G$, denoted by $\iota(G,\mathcal{F})$, is the size of a smallest subset $D$ of the vertex set $V(G)$ such that the closed neighbourhood of $D$ intersects the vertex sets of the $\mathcal{F}$-graphs contained by $G$ (equivalently, $G - N[D]$ contains no $\mathcal{F}$-graph). Thus, $\iota(G,\{K_1\})$ is the domination number of $G$. The second author showed that if $\mathcal{F}$ is the set of cycles and $G$ is a connected $n$-vertex graph that is not a triangle, then $\iota(G,\mathcal{F}) \leq \left \lfloor \frac{n}{4} \right \rfloor$. This bound is attainable for every $n$ and solved a problem of Caro and Hansberg. A question that arises immediately is how smaller an upper bound can be if $\mathcal{F} = \{C_k\}$ for some $k \geq 3$, where $C_k$ is a cycle of length $k$. The problem is to determine the smallest real number $c_k$ (if it exists) such that for some finite set $\mathcal{E}_k$ of graphs, $\iota(G, \{C_k\}) \leq c_k |V(G)|$ for every connected graph $G$ that is not an $\mathcal{E}_k$-graph. The above-mentioned result yields $c_3 = \frac{1}{4}$ and $\mathcal{E}_3 = \{C_3\}$. The second author also showed that if $k \geq 5$ and $c_k$ exists, then $c_k \geq \frac{2}{2k + 1}$. We prove that $c_4 = \frac{1}{5}$ and determine $\mathcal{E}_4$, which consists of three $4$-vertex graphs and six $9$-vertex graphs. The $9$-vertex graphs in $\mathcal{E}_4$ were fully determined by means of a computer program. A method that has the potential of yielding similar results is introduced.Comment: 15 pages, 1 figure. arXiv admin note: text overlap with arXiv:2110.0377

Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths

When can a plane graph with prescribed edge lengths and prescribed angles (from among $\{0,180^\circ, 360^\circ$\}) be folded flat to lie in an infinitesimally thin line, without crossings? This problem generalizes the classic theory of single-vertex flat origami with prescribed mountain-valley assignment, which corresponds to the case of a cycle graph. We characterize such flat-foldable plane graphs by two obviously necessary but also sufficient conditions, proving a conjecture made in 2001: the angles at each vertex should sum to $360^\circ$, and every face of the graph must itself be flat foldable. This characterization leads to a linear-time algorithm for testing flat foldability of plane graphs with prescribed edge lengths and angles, and a polynomial-time algorithm for counting the number of distinct folded states.Comment: 21 pages, 10 figure

NC Algorithms for Computing a Perfect Matching and a Maximum Flow in One-Crossing-Minor-Free Graphs

In 1988, Vazirani gave an NC algorithm for computing the number of perfect matchings in $K_{3,3}$-minor-free graphs by building on Kasteleyn's scheme for planar graphs, and stated that this "opens up the possibility of obtaining an NC algorithm for finding a perfect matching in $K_{3,3}$-free graphs." In this paper, we finally settle this 30-year-old open problem. Building on recent NC algorithms for planar and bounded-genus perfect matching by Anari and Vazirani and later by Sankowski, we obtain NC algorithms for perfect matching in any minor-closed graph family that forbids a one-crossing graph. This family includes several well-studied graph families including the $K_{3,3}$-minor-free graphs and $K_5$-minor-free graphs. Graphs in these families not only have unbounded genus, but can have genus as high as $O(n)$. Our method applies as well to several other problems related to perfect matching. In particular, we obtain NC algorithms for the following problems in any family of graphs (or networks) with a one-crossing forbidden minor: $\bullet$ Determining whether a given graph has a perfect matching and if so, finding one. $\bullet$ Finding a minimum weight perfect matching in the graph, assuming that the edge weights are polynomially bounded. $\bullet$ Finding a maximum $st$-flow in the network, with arbitrary capacities. The main new idea enabling our results is the definition and use of matching-mimicking networks, small replacement networks that behave the same, with respect to matching problems involving a fixed set of terminals, as the larger network they replace.Comment: 21 pages, 6 figure

Minor-Obstructions for Apex-Pseudoforests

A graph is called a pseudoforest if none of its connected components contains more than one cycle. A graph is an apex-pseudoforest if it can become a pseudoforest by removing one of its vertices. We identify 33 graphs that form the minor-obstruction set of the class of apex-pseudoforests, i.e., the set of all minor-minimal graphs that are not apex-pseudoforests