Algorithms for Maximum Internal Spanning Tree Problem for Some Graph Classes ¹¹1The third author is supported by an NSF Graduate Research Fellowship under Grant No. DGE-1650044.

The Maximum Internal Spanning Tree (MIST) problem is a degree based spanning tree optimization problem, in which we ask to find a spanning tree of a given graph such that the number of vertices of degree at least two is maximized. The MIST problem is motivated by telecommunication network design [22]. We also believe that MIST problem has its own theoretical importance as it is a generalization of the Hamiltonian path problem, a known NP-complete problem [5]. The Hamiltonian path problem remains NP-complete for chordal graphs and chordal bipartite graphs [10, 18]. Hence, we also do not expect polynomial time algorithms for the MIST problem in chordal graphs and chordal bipartite graphs.

The dual problem to MIST, the Minimum Leaves Spanning Tree (MLST) problem asks to find a spanning tree with minimum number of leaves for a given graph. The MLST problem cannot be approximated within any constant factor unless P=NP [17]. Unlike MLST, several constant factor approximation algorithms have been proposed for the MIST problem in literature. In $2003$, Prieto et al. [20] gave a $2$-approximation algorithm for the MIST problem whose running time was later improved by Salamon et al. in 2008 [23]. Salamon also gave approximation algorithms for claw-free and cubic graphs with approximation factors $\frac{3}{2}$ and $\frac{6}{5}$ respectively [23]. In 2009, Salamon [21] gave a $\frac{7}{4}$ -approximation algorithm for graphs with no pendant vertices and later, in 2015, Knauer et al. [9] showed that a simplified and faster version of Salamon’s algorithm yields a $\frac{5}{3}$ -approximation algorithm even on general graphs. In 2014, Li et al. proposed a $\frac{3}{2}$-approximation algorithm using a different approach for general undirected graphs and improved this ratio to $\frac{4}{3}$ for graphs without leaves [14]. Li et al. gave a $\frac{3}{2}$ -approximation algorithm for general graphs using depth-5 local search [13]. In 2018, Chen et al. presented a $\frac{17}{13}$ -approximation algorithm which is the best approximation factor till now [2]. Several FPT-algorithms have also been designed for the MIST problem where the considered parameter is the solution size [20, 13, 3, 4, 1, 12].

For finding efficient algorithms for the MIST problem, it is often useful to reduce the MIST problem to the path cover problem. A path cover $P$ of a graph is a spanning subgraph such that every component of $P$ is a path. A path cover with maximum number of edges is called an optimal path cover of $G$. If $P^{*}$ denotes an optimal path cover of a graph, then number of edges in $P^{*}$ is denoted by $|E(P^{*})|$. In 2018, Li et al. proposed a polynomial time algorithm for the MIST problem in interval graphs [15]. They also proved that number of internal vertices in any MIST of any graph $G$ is at most $|E(P^{*})|-1$, where $P^{*}$ is an optimal path cover of $G$. We will observe that number of internal vertices in any MIST of a chain graph is either $|E(P^{*})|-1$ or $|E(P^{*})|-2$ and is $|E(P^{*})|-1$ for cographs. For bipartite permutation, block and cactus graphs, we prove that there is no constant $k$ such that $|E(P^{*})|-k$ is the lower bound on the number of internal vertices in any MIST of such graphs. We also propose linear-time algorithms for the MIST problem in bipartite permutation graphs, block graphs, cactus graphs and cographs. A hierarchy relationship between these classes of graphs is shown in Fig. 1.

The structure of the paper is as follows. In Section 2, we give some basic definitions and notations used in the paper. In Section 3, we discuss MIST problem for block graphs and cactus graphs and provide linear-time algorithms for both these graph classes. In Section 4, we prove that MIST of cographs can be computed in linear-time by providing an algorithm. In Section 5, we present a linear-time algorithm to find a MIST of an arbitrary bipartite permutation graph. In Section 6, we prove a bound for chain graphs regarding number of internal vertices in its MIST. Finally, Section 7 concludes the paper.

Let $G=(V,E)$ be a graph. In this paper, we only consider simple, undirected and connected graphs. For a vertex $u\in G,d_{G}(u)$ denotes the degree of $u$ in $G$ and $N_{G}(u)$ denotes the neighborhood of $u$ in $G$. When there is no ambiguity regarding the graph $G$, we simply use $d(u)$ and $N(u)$, to represent the degree of $u$ and neighborhood of $u$, respectively. A vertex $u$ in $V$ is called pendant if $d(u)=1$. The set of pendant vertices in $G$ is denoted by $p(G)$. The vertex adjacent to a pendant vertex $u$ is called the support vertex of $u$, and is denoted by $S(u)$. A vertex $u\in V(G)$ is called internal, if $u$ is not pendant, that is, $d(u)\geq 2$. Let $I(G)$ denotes the set of internal vertices in $G$, and $i(G)=|I(G)|$. For a set $A\subseteq V$ and a spanning tree $T$ of $G$, we define $i_{T}(A)=|{I(T)\cap A}|$.

For vertices $x$ and $y$, we denote an edge between $x$ and $y$ by $xy$. For a subset $S$ of $V(G)$, $G-S$ denotes the subgraph of $G$ obtained after removing vertices of $S$ and edges incident on these vertices from $G$. If $S=\{v\}$, then we simply write $G-v$ for $G-S$. A vertex $v$ of a graph $G$ is called a cut vertex if $G-v$ is disconnected.

Throughout this paper, $n$ denotes the number of vertices and $m$ denotes the number of edges in $G$. A graph $G$ is said to be bipartite if $V$ can be partitioned into two disjoint sets $X$ and $Y$ such that every edge of $G$ joins a vertex in $X$ to a vertex in $Y$. Such a partition $(X,Y)$ of $V$ is called a bipartition. A bipartite graph with bipartition $(X,Y)$ of $V$ is denoted by $G=(X,Y,E)$. For a set $S\subseteq V$, an induced subgraph is the graph whose vertex set is $S$ and edge set consists of all the edges in $E$ that have both the endpoints in $S$, and is denoted by $G[S]$. If $G[C],C\subseteq V$, is a complete subgraph of $G$, then $C$ is called a clique of $G$.

A subgraph of $G$ is called a spanning subgraph if it contains all the vertices of $G$. A path cover $P$ of a graph is a spanning subgraph such that every component of $P$ is a path. A path cover is an optimal path cover if it has the maximum number of edges. A spanning subgraph of $G$ which is also a tree is called a spanning tree of $G$. A spanning tree is called a maximum internal spanning tree(MIST) if it contains the maximum number of internal vertices among all the spanning trees of $G$. For a graph $G$, the number of internal vertices in any MIST of $G$ is denoted by $Opt(G)$.

Now we state a useful theorem which gives an upper bound on the number of internal vertices in a MIST with respect to the graph’s optimal path cover.

Note that the vertices which are pendant in $G$ itself, will be pendant in any MIST of $G$. Hence, we have the following observation.

Suppose $G$ is not a tree and $u\in V(G)$ is adjacent to $k$ pendant vertices, say $a_{1},\ldots,a_{k}$. Let $G^{\prime}=G-\{a_{2},\ldots,a_{k}\}$. Then based on above observation, the number of internal vertices in a MIST of $G$ will be same as the number of internal vertices in any MIST of $G^{\prime}$. It is also easy to obtain a MIST of $G$ from any MIST of $G^{\prime}$. Hence, throughout this work, we assume that every vertex has at most one pendant vertex adjacent with it.

Below, we give another result regarding the number of pendant vertices in a spanning tree of a bipartite graph. Note that, if we have $\alpha$ number of internal vertices in a spanning tree of $G$ from one partite set, then at least $\alpha+1$ vertices must be present in the neighborhood of these $\alpha$ vertices, which lie in the other partite set of the bipartite graph $G$.

In this section, we discuss the MIST problem for block graphs and cactus graphs. We will show that the MIST problem can be solved in linear-time for both classes of graphs. Block and cactus graphs will also provide our first family of examples in which $Opt(G)$ cannot be lower bounded in terms of $|E(P^{*})|-k$ where $P^{*}$ is an optimal path cover of $G$ and $k$ is some constant.

A block of a graph $G$ is a maximal connected subgraph with no cut vertices. Note that a block of $G$ is either an edge or a 2-connected subgraph. The set of blocks of a graph is called the block decomposition of $G$ and is denoted by $B(G)$. Let $B_{0}\in B(G)$ and $u,v$ be two vertices belonging to $B_{0}$, then a path between $u$ and $v$, which contains all the vertices of the block $B_{0}$, is called a spanning path between $u$ and $v$ in $B_{0}$. We say a block $B$ is good if there exists distinct $u,v\in V(B)$ such that both $u$ and $v$ are cut vertices of $G$ and $B$ has a spanning path between $u$ and $v$. A block is said to be bad otherwise. Let $\text{Bad}(G)$ denote the set of bad blocks of $G$.

A block graph is a graph in which every block is a clique. If a block graph $G$ contains only one block then $G$ is a complete graph. A block graph is said to be nontrivial if it contains at least two blocks. Note that a trivial block has a Hamiltonian path. Thus for the remainder of the section we only consider nontrivial block graphs.

Let $G$ be a nontrivial block graph. Bad blocks of $G$ have another characterization which we state as the Observation 3.1.

A graph $G$ is a cactus graph if every block of $G$ is either a cycle or an edge. If a cactus graph $G$ contains only one block then $G$ is either a cycle or an edge and in that case finding a MIST of $G$ is trivial. Again, a cactus graph is said to be nontrivial if it contains at least two blocks and now we only consider nontrivial cactus graphs.

Let $G$ be a nontrivial cactus graph. A block of $G$ is called an end block of $G$ if it contains exactly one cut vertex of $G$. Note that an end block of a cactus graph $G$ is also a bad block of $G$. Bad blocks of a cactus graph $G$ have another characterization which we state in the following observation.

If $B_{i}$ and $B_{j}$ are two blocks of a block/cactus graph $G$ and $V(B_{i})\cap V(B_{j})\neq\emptyset$, then $V(B_{i})\cap V(B_{j})=1$ and the vertex $x\in V(B_{i})\cap V(B_{j})$ is a cut vertex of $G$. Let $T$ be a MIST of a nontrivial block/cactus graph $G$. Below, we state two observations which hold true for both block and cactus graphs.

Recall that block decomposition of a graph $G$ is the set of blocks of $G$. It can be computed in $O(n)$ time using the following approach. Let $b$ be a cut vertex of a block/cactus graph $G$ and $G_{1},G_{2},\ldots,G_{t}$ be the connected components of the graph $G-b$. Let $H_{i}$ denotes the subgraph $G[V(G_{i})\cup\{b\}]$, for each $1\leq i\leq t$. We call $H_{1},H_{2},\ldots,H_{t}$ the $b$-components of $G$. The block decomposition of a block/cactus graph can be found by recursively choosing a cut vertex $b$ and computing the $b$-components.

In this section, we discuss the MIST problem for cographs. The complement-reducible graphs or cographs have been discovered independently by several authors since the 1970s [24, 8]. In the literature, the cographs are also known as $P_{4}$-free graphs, $D^{*}$-graphs, Hereditary Dacey graphs and $2$-parity graphs. The class of cographs is defined recursively as follows:

• •

  A single-vertex graph is a cograph;

• •

  If $G$ is a cograph, then its complement $\overline{G}$ is also a cograph;

• •

  If $G$ and $H$ are cographs, then their disjoint union is also a cograph.

Cographs admit a rooted tree representation. This tree is called cotree of a cograph $G$, denoted $T(G)$. The cotree of a cograph rooted at a node $r$ possesses the following properties.

1. 1.

   Every internal vertex except $r$ has at least two children. Furthermore, $r$ has exactly one child if and only if the underlying cograph $G$ is disconnected.

2. 2.

   The internal nodes are labeled by either $0$ ($0$-nodes) or $1$ ($1$-nodes) in such a way that the root is always a $1$-node, and $1$-nodes and $0$-nodes alternate along every path in $T(G)$ starting at the root.

3. 3.

   The leaves of $T(G)$ are precisely the vertices of $G$, such that vertices $x$ and $y$ are adjacent in $G$ if, and only if, the lowest common ancestor of $x$ and $y$ in $T(G)$ is a $1$-node.

Fig. 3 illustrates a cograph $G$ along with its cotree $T(G)$.

According to [16], the cotree of any cograph $G$ can be preprocessed such that it is a binary tree. So, we may assume that $T(G)$ is a binary tree rooted at a vertex $r$. The set of leaves of the left subtree of $T(G)$ is denoted by $L(r_{left})$ and the set of leaves of the right subtree of $T(G)$ is denoted by $L(r_{right})$.

Note that every leaf of $T(G)$ represents a vertex of the graph $G$. If we consider one vertex from $L(r_{left})$ and one vertex from $L(r_{right})$ then their least common ancestor is the root node. As the root node is always a $1$-node, we have the following observation.

Recall that a path cover $P$ of a graph $G$ is a spanning subgraph such that every component of $P$ is a path. A path cover is an optimal path cover if it has the maximum number of edges. [16] gave a linear-time algorithm to compute an optimal path cover of a cograph $G$. The optimal path cover $P^{*}$ constructed in [16] is one of the following type:

• •

  The path cover $P^{*}$ contains a single path component which is a Hamiltonian path of $G$.
  

• •

  The path cover $P^{*}$ contains at least two path components. In this case, there exists exactly one path $p$ in $P^{*}$ which contains vertices from both the sets $L(r_{left})$ and $L(r_{right})$ and all other paths in $P^{*}$ contain vertices from $L(r_{right})$ only. Fig. 4 illustrates this case.

Algorithm 1 uses the optimal path cover constructed from [16] to compute a MIST of a cograph $G$.

Note that by Theorem 2.1 we have $Opt(G)\leq|{E(P^{*})}|-1$ for an optimal path cover $P^{*}$. Below, we give a theorem which implies that Algorithm 1 also outputs a spanning tree which attains this upper bound.

Note that step 1 of Algorithm 1 can be performed in linear-time [16]. Furthermore, note that the construction of $T$ in Theorem 4.1 is also linear-time. Therefore Algorithm 1 outputs a MIST of $G$ in linear-time.

In this section, we discuss the MIST problem for bipartite permutation graphs. A graph $G=(V,E)$ with $V=\{v_{1},v_{2},\ldots,v_{n}\}$ is said to be a permutation graph if there is a permutation $\sigma$ over $\{1,2,\ldots,n\}$ such that $v_{i}v_{j}\in E$ if and only if $(i-j)(\sigma(i)-\sigma(j))<0$. Intuitively, each vertex $v$ in a permutation graph corresponds to a line segment $l_{v}$ joining two points on two parallel lines $L_{1}$ and $L_{2}$, which is called a line representation. Then, two vertices $v$ and $u$ are adjacent if and only if the corresponding line segments $l_{v}$ and $l_{u}$ are crossing. Vertex indices give the ordering of the points on $L_{1}$, and the permutation of the indices gives the ordering of the points on $L_{2}$. When a permutation graph is bipartite, it is said to be a bipartite permutation graph.

A strong ordering $(<_{X},<_{Y})$ of a bipartite graph $G=(X,Y,E)$ consists of an ordering $<_{X}$ of $X$ and an ordering $<_{Y}$ of $Y$, such that for all edges $ab,a^{\prime}b^{\prime}$ with $a,a^{\prime}\in X$ and $b,b^{\prime}\in Y$: if $a<_{X}a^{\prime}$ and $b^{\prime}<_{Y}b,$ then $ab^{\prime}$ and $a^{\prime}b$ are edges in $G$. An ordering $<_{X}$ of $X$ has the adjacency property if, for every vertex in $Y$, its neighbors in $X$ are consecutive in $<_{X}$. The ordering $<_{X}$ has the enclosure property if, for every pair of vertices $y,y^{\prime}$ of $Y$ with $N(y)\subseteq N(y^{\prime})$, the vertices of $N(y^{\prime})\setminus N(y)$ appear consecutively in $<_{X}$. These properties are useful for characterizing bipartite permutation graphs.

[7] proved that a bipartite graph is a bipartite permutation graph if and only if it admits a strong ordering. Furthermore if we assume that the graph is connected, then we can say more.

Throughout this section, $G=(X,Y,E)$ denotes a connected bipartite permutation graph. A strong ordering of a bipartite permutation graph can be computed in linear-time [25]. Let $(<_{X},<_{Y})$ be a strong ordering of $G$, where $<_{X}=(x_{1},x_{2},\ldots,x_{n_{1}})$ and $<_{Y}=(y_{1},y_{2},\ldots,y_{n_{2}})$. We write strong ordering of vertices of $G$ as $(<_{X},<_{Y})=(x_{1},x_{2},\ldots,x_{n_{1}},y_{1},y_{2},\ldots,y_{n_{2}})$. For $u,v\in V(G)$, we write $u<_{X}v$ if $u,v\in X$ and $u$ appears before $v$ in the strong ordering; we define $u<_{Y}v$ in a similar manner. We write $x_{i}<x_{j}$ (or, $y_{i}<y_{j}$) when $i<j$. For vertices $u,v$ of $G$, $u\leq v$ denotes either $u<_{X}v$, $u<_{Y}v$, or $u=v$ holds.

Since each vertex of $G$ satisfies the adjacency property, the neighborhood of any vertex consists of consecutive vertices in the strong ordering. We define the first neighbor of a vertex as the vertex with minimum index in its neighborhood and the last neighbor of a vertex as the vertex with maximum index in its neighborhood. We notate the first and last neighbors of a vertex $u$ as $f(u)$ and $l(u)$ respectively. Combining the above statements for a bipartite permutation graph $G$ with its strong ordering $(<_{X},<_{Y})$, $G$ has the following properties [11]:

1. 1.

   For any vertex of $G$, its neighborhood consists of consecutive vertices in $<_{X}$ or $<_{Y}$ .

2. 2.

   For any pair of vertices $u,v$ from $X$ or $Y$, if $u<v$ then $f(u)\leq f(v)$ and $l(u)\leq l(v)$.

Now, we define some terminology which we require for the remainder of this section. A vertex $x_{i}\in X,(1\leq i\leq n_{1})$ with $l(x_{i})=y_{j}$ is of type 1 if $j\geq i$. A vertex $y_{i}\in Y,(1\leq i\leq n_{2})$ with $l(y_{i})=x_{j}$ is of type 1 if $j\geq i+1$. Similarly, a vertex $x_{i}\in X,(1\leq i\leq n_{1})$ with $l(x_{i})=y_{j}$ is of type 2 if $j\geq i+1$ and a vertex $y_{i}\in Y,(1\leq i\leq n_{2})$ with $l(y_{i})=x_{j}$ is of type 2 if $j\geq i$. Note that a type 2 vertex $x\in X$ is also a type 1 vertex but the converse may not be true. Furthermore, a type 1 vertex $y\in Y$ is also a type 2 vertex. Characterizing the vertices in this way is an important distinction for our algorithm. We now prove two important lemmas which will be used to prove the correctness of Algorithm 2.