A bipartite graph is chordal bipartite if every cycle of length at least six has a chord in it. M$\ddot{\rm u}$ller \cite {muller1996Hamiltonian} has shown that the Hamiltonian cycle problem is NP-complete on chordal bipartite graphs by presenting a polynomial-time reduction from the satisfiability problem. The microscopic view of the reduction instances reveals that the instances are $P_9$-free chordal bipartite graphs, and hence the status of Hamiltonicity in $P_8$-free chordal bipartite graphs is open. In this paper, we identify the first non-trivial subclass of $P_8$-free chordal bipartite graphs which is $P_5$-free chordal bipartite graphs, and present structural and algorithmic results on $P_5$-free chordal bipartite graphs. We investigate the structure of $P_5$-free chordal bipartite graphs and show that these graphs have a {\em Nested Neighborhood Ordering (NNO)}, a special ordering among its vertices. Further, using this ordering, we present polynomial-time algorithms for classical problems such as the Hamiltonian cycle (path), also the variants and generalizations of the Hamiltonian cycle (path) problem. We also obtain polynomial-time algorithms for treewidth (pathwidth), and minimum fill-in in $P_5$-free chordal bipartite graph. We also present some results on complement graphs of $P_5$-free chordal bipartite graphs.
We present an algorithm for determining whether a bipartite graph $G$ is 2-chordal (formerly doubly chordal bipartite). At its core this algorithm is an extension of the existing efficient algorithm for determining whether a graph is chordal bipartite. We then introduce the notion of $k$-chordal bipartite graphs and show by inductive means that a slight modification of our algorithm is sufficient to detect this property. We show that there are no nontrivial $k$-chordal bipartite graphs for $k \geq 4$ and that both the 2-chordal bipartite and 3-chordal bipartite problem are contained within com…
The mirror (or bipartite complement) mir(B) of a bipartite graph B=(X,Y,E) has the same color classes X and Y as B, and two vertices x in X and y in Y are adjacent in mir(B) if and only if xy is not in E. A bipartite graph is chordal bipartite if none of its induced subgraphs is a chordless cycle with at least six vertices. In this paper, we deal with chordal bipartite graphs whose mirror is chordal bipartite as well; we call these graphs auto-chordal bipartite graphs (ACB graphs for short). We describe the relationship to some known graph classes such as interval and strongly chordal graphs a…
For a connected graph, the Hamiltonian cycle (path) is a simple cycle (path) that spans all the vertices in the graph. It is known from \cite{muller,garey} that HAMILTONIAN CYCLE (PATH) are NP-complete in general graphs and chordal bipartite graphs. A convex bipartite graph $G$ with bipartition $(X,Y)$ and an ordering $X=(x_1,\ldots,x_n)$, is a bipartite graph such that for each $y \in Y$, the neighborhood of $y$ in $X$ appears consecutively. $G$ is said to have convexity with respect to $X$. Further, convex bipartite graphs are a subclass of chordal bipartite graphs. In this paper, we present…
We define strongly chordal digraphs, which generalize strongly chordal graphs and chordal bipartite graphs, and are included in the class of chordal digraphs. They correspond to square 0,1 matrices that admit a simultaneous row and column permutation avoiding the Γ matrix. In general, it is not clear if these digraphs can be recognized in polynomial time, and we focus on symmetric digraphs (i.e., graphs with possible loops), tournaments with possible loops, and balanced digraphs. In each of these cases we give a polynomial-time recognition algorithm and a forbidden induced subgraph characteriz…
A bipartite graph is chordal bipartite if every cycle of length at least 6 has a chord in it. In this paper, we investigate the structure of $P_5$-free chordal bipartite graphs and show that these graphs have a Nested Neighborhood Ordering, a special ordering among its vertices. Further, using this ordering, we present polynomial-time algorithms for classical problems such as Hamiltonian cycle (path) and longest path. Two variants of Hamiltonian path include Steiner path and minimum leaf spanning tree, and we obtain polynomial-time algorithms for these problems as well restricted to $P_5$-free…
The class of $k$-star caterpillar convex bipartite graphs generalizes the class of convex bipartite graphs. For a bipartite graph with partitions $X$ and $Y$, we associate a $k$-star caterpillar on $X$ such that for each vertex in $Y$, its neighborhood induces a tree. The $k$-star caterpillar on $X$ is imaginary and if the imaginary structure is a path ($0$-star caterpillar), then it is the class of convex bipartite graphs. The minimum Steiner tree problem (STREE) is defined as follows: given a connected graph $G=(V,E)$ and a subset of vertices $R \subseteq V(G)$, the objective is to find a mi…
A vertex set $D$ in a finite undirected graph $G$ is an {\em efficient dominating set} (\emph{e.d.s.}\ for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \emph{Efficient Domination} (ED) problem, which asks for the existence of an e.d.s.\ in $G$, is known to be \NP-complete for chordal bipartite graphs as well as for $P_7$-free graphs, and even for very restricted $H$-free bipartite graph classes such as for $K_{1,4}$-free bipartite graphs as well as for $C_4$-free bipartite graphs while it is solvable in polynomial time for $P_8$-free bipartite graphs as w…
Let G=(V,E) be a connected graph. A set U subseteq V is convex if G[U] is connected and all vertices of V\U have at most one neighbor in U. Let sigma(W) denote the unique smallest convex set that contains W subseteq V. Two players play the following game. Consider a convex set U and call it the `playground.' Initially, U = emptyset. When U=V, the player to move loses the game. Otherwise, that player chooses a vertex x in V\U which is at distance at most two from U. The effect of the move is that the playground U changes into sigma(U cup {x}) and the opponent is presented with this new playgrou…
Let $B=(X,Y,E)$ be a bipartite graph. A half-square of $B$ has one color class of $B$ as vertex set, say $X$; two vertices are adjacent whenever they have a common neighbor in $Y$. Every planar graph is a half-square of a planar bipartite graph, namely of its subdivision. Until recently, only half-squares of planar bipartite graphs, also known as map graphs (Chen, Grigni and Papadimitriou [STOC 1998, J. ACM 2002]), have been investigated, and the most discussed problem is whether it is possible to recognize these graphs faster and simpler than Thorup's $O(n^{120})$-time algorithm (Thorup [FOCS…
We characterize clique trees of a chordal graph in their relation to simplicial vertices and perfect sequences of maximal cliques. We investigate boundary cliques defined by Shibata and clarify their relation to endpoints of clique trees. Next we define a symmetric binary relation between the set of clique trees and the set of perfect sequences of maximal cliques. We describe the relation as a bipartite graph and prove that the bipartite graph is always connected. Lastly we consider to characterize chordal graphs from the aspect of non-uniqueness of clique trees.
Strictly Chordality-k graphs (SC_k graphs) are graphs which are either cycle free or every induced cycle is exactly k, for some fixed k, k \geq 3. Note that k = 3 and k = 4 are precisely the Chordal graphs and Chordal Bipartite graphs, respectively. In this paper, we initiate a structural and an algorithmic study of SCk, k \geq 5 graphs.
The biclique partition number $(\text{bp})$ of a graph $G$ is referred to as the least number of complete bipartite (biclique) subgraphs that are required to cover the edges of the graph exactly once. In this paper, we show that the biclique partition number ($\text{bp}$) of a co-chordal (complementary graph of chordal) graph $G = (V, E)$ is less than the number of maximal cliques ($\text{mc}$) of its complementary graph: a chordal graph $G^c = (V, E^c)$. We first provide a general framework of the ``divide and conquer" heuristic of finding minimum biclique partitions of co-chordal graphs base…
In a graph $G=(V,E)$ with no isolated vertex, a dominating set $D \subseteq V$, is called a semitotal dominating set if for every vertex $u \in D$ there is another vertex $v \in D$, such that distance between $u$ and $v$ is at most two in $G$. Given a graph $G=(V,E)$ without isolated vertices, the Minimum Semitotal Domination problem is to find a minimum cardinality semitotal dominating set of $G$. The semitotal domination number, denoted by $γ_{t2}(G)$, is the minimum cardinality of a semitotal dominating set of $G$. The decision version of the problem remains NP-complete even when restricted…
List k-Coloring (Li k-Col) is the decision problem asking if a given graph admits a proper coloring compatible with a given list assignment to its vertices with colors in {1,2,..,k}. The problem is known to be NP-hard even for k=3 within the class of 3-regular planar bipartite graphs and for k=4 within the class of chordal bipartite graphs. In 2015, Huang, Johnson and Paulusma asked for the complexity of Li 3-Col in the class of chordal bipartite graphs. In this paper we give a partial answer to this question by showing that Li k-Col is polynomial in the class of convex bipartite graphs. We sh…
\emph{Strictly Chordality-$k$ graphs ($SC_k$)} are graphs which are either cycle-free or every induced cycle is of length exactly $k, k \geq 3$. Strictly chordality-3 and strictly chordality-4 graphs are well known chordal and chordal bipartite graphs, respectively. For $k\geq 5$, the study has been recently initiated in \cite{sadagopan} and various structural and algorithmic results are reported. In this paper, we show that maximum independent set (MIS), minimum vertex cover, minimum dominating set, feedback vertex set (FVS), odd cycle transversal (OCT), even cycle transversal (ECT) and Stein…
For an even integer t \geq 2, the Matchings Connecivity matrix H_t is a matrix that has rows and columns both labeled by all perfect matchings of the complete graph K_t on t vertices; an entry H_t[M_1,M_2] is 1 if M_1\cup M_2 is a Hamiltonian cycle and 0 otherwise. Motivated by the computational study of the Hamiltonicity problem, we present three results on the structure of H_t: We first show that H_t has rank at most 2^{t/2-1} over GF(2) via an appropriate factorization that explicitly provides families of matchings X_t forming bases for H_t. Second, we show how to quickly change representat…
A graph $G$ is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function $w$ is defined on its vertices. Then $G$ is $w$-well-covered if all maximal independent sets are of the same weight. For every graph $G$, the set of weight functions $w$ such that $G$ is $w$-well-covered is a vector space, denoted $WCW(G)$. Let $B$ be a complete bipartite induced subgraph of $G$ on vertex sets of bipartition $B_{X}$ and $B_{Y}$. Then $B$ is generating if there exists an independent set $S$ such that $S \cup B_{X}$ and $S \cup B_{Y}$ are both maximal indepe…
Barnette's Conjecture claims that all cubic, 3-connected, planar, bipartite graphs are Hamiltonian. We give a translation of this conjecture into the matching-theoretic setting. This allows us to relax the requirement of planarity to give the equivalent conjecture that all cubic, 3-connected, Pfaffian, bipartite graphs are Hamiltonian. A graph, other than the path of length three, is a brace if it is bipartite and any two disjoint edges are part of a perfect matching. Our perspective allows us to observe that Barnette's Conjecture can be reduced to cubic, planar braces. We show a similar reduc…
Let $G=(V, E)$ be a graph, where $V$ and $E$ are the vertex and edge sets, respectively. For two disjoint subsets $A$ and $B$ of $V$, we say $A$ \textit{dominates} $B$ if every vertex of $B$ is adjacent to at least one vertex of $A$ in $G$. A vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{transitive $k$-partition} if $V_i$ dominates $V_j$ for all $i,j$, where $1\leq i<j\leq k$. The maximum integer $k$ for which the above partition exists is called \emph{transitivity} of $G$ and it is denoted by $Tr(G)$. The \textsc{Maximum Transitivity Problem} is to find a transitive…
Lexicographic Breadth First Search (LBFS) is one of fundamental graph search algorithms that has numerous applications, including recognition of graph classes, computation of graph parameters, and detection of certain graph structures. The well-known result of Rose, Tarjan and Lueker on the end-vertices of LBFS of chordal graphs has tempted researchers to study the end-vertices of LBFS of various classes of graphs, including chordal graphs, split graphs, interval graphs, and asteroidal triple-free (AT-free) graphs. In this paper we study the end-vertices of LBFS of bipartite graphs. We show th…
The class of Bipartite Distance Hereditary (BDH) graphs is the intersection between bipartite domino-free and chordal bipartite graphs.\ Graphs in both the latter classes have linearly many maximal bicliques, implying the existence of polynomial-time algorithms for computing the associated Galois lattice.\ Such a lattice can indeed be built in $O(m\times n)$ worst case-time for a domino-free graph with $m$ edges and $n$ vertices.\ In this paper we give a sharp estimate on the number of the maximal bicliques of BDH graphs and exploit such result to give an $O(m)$ worst case time algorithm for c…
This research contains as a main result the prove that every Chordal $B_1$-EPG graph is simultaneously in the graph classes VPT and EPT. In addition, we describe structures that must be present in any $B_1$-EPG graph which does not admit a Helly-$B_1$-EPG representation. In particular, this paper presents some features of non-trivial families of graphs properly contained in Helly-$B_1$ EPG, namely Bipartite, Block, Cactus and Line of Bipartite graphs.
Let $G = (V,E)$ be a simple, undirected and connected graph. A connected dominating set $S \subseteq V$ is a secure connected dominating set of $G$, if for each $ u \in V\setminus S$, there exists $v\in S$ such that $(u,v) \in E$ and the set $(S \setminus \{ v \}) \cup \{ u \} $ is a connected dominating set of $G$. The minimum size of a secure connected dominating set of $G$ denoted by $ γ_{sc} (G)$, is called the secure connected domination number of $G$. Given a graph $ G$ and a positive integer $ k,$ the Secure Connected Domination (SCDM) problem is to check whether $ G $ has a secure conn…
For a simple graph $G=(V,E)$ without any isolated vertex, a cosecure dominating set $D$ of $G$ satisfies the following two properties (i) $S$ is a dominating set of $G$, (ii) for every vertex $v \in S$ there exists a vertex $u \in V \setminus S$ such that $uv \in E$ and $(S \setminus \{v\}) \cup \{u\}$ is a dominating set of $G$. The minimum cardinality of a cosecure dominating set of $G$ is called cosecure domination number of $G$ and is denoted by $γ_{cs}(G)$. The Minimum Cosecure Domination problem is to find a cosecure dominating set of a graph $G$ of cardinality $γ_{cs}(G)$. The decision …
Matrix partition problems generalize a number of natural graph partition problems, and have been studied for several standard graph classes. We prove that each matrix partition problem has only finitely many minimal obstructions for split graphs. Previously such a result was only known for the class of cographs. (In particular, there are matrix partition problems which have infinitely many minimal chordal obstructions.) We provide (close) upper and lower bounds on the maximum size of a minimal split obstruction. This shows for the first time that some matrices have exponential-sized minimal ob…
A vertex set $D$ in a finite undirected graph $G$ is an {\em efficient dominating set} (\emph{e.d.s.}\ for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \emph{Efficient Domination} (ED) problem, which asks for the existence of an e.d.s.\ in $G$, is \NP-complete for various $H$-free bipartite graphs, e.g., Lu and Tang showed that ED is \NP-complete for chordal bipartite graphs and for planar bipartite graphs; actually, ED is \NP-complete even for planar bipartite graphs with vertex degree at most 3 and girth at least $g$ for every fixed $g$. Thus, ED is \NP…
A vertex set $D$ in a finite undirected graph $G$ is an {\em efficient dominating set} (e.d.s.\ for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \emph{Efficient Domination} (ED) problem, which asks for the existence of an e.d.s.\ in $G$, is \NP-complete for various $H$-free bipartite graphs, e.g., Lu and Tang showed that ED is \NP-complete for chordal bipartite graphs and for planar bipartite graphs; actually, ED is \NP-complete even for planar bipartite graphs with vertex degree at most 3 and girth at least $g$ for every fixed $g$. Thus, ED is \NP-comple…
Eternal vertex cover problem is a variant of the classical vertex cover problem modeled as a two player attacker-defender game. Computing eternal vertex cover number of graphs is known to be NP-hard in general and the complexity status of the problem for bipartite graphs is open. There is a quadratic complexity algorithm known for this problem for chordal graphs. Maximal outerplanar graphs forms a subclass of chordal graphs, for which no algorithm of sub-quadratic time complexity is known. In this paper, we obtain a recursive algorithm of linear time for computing eternal vertex cover number o…
A vertex set $D$ in a finite undirected graph $G$ is an {\em efficient dominating set} (\emph{e.d.s.}\ for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \emph{Efficient Domination} (ED) problem, which asks for the existence of an e.d.s.\ in $G$, is known to be \NP-complete even for very restricted $H$-free graph classes such as for $2P_3$-free chordal graphs while it is solvable in polynomial time for $P_6$-free graphs. Here we focus on $H$-free bipartite graphs: We show that (weighted) ED can be solved in polynomial time for $H$-free bipartite graphs when…
We study the class of 1-perfectly orientable graphs, that is, graphs having an orientation in which every out-neighborhood induces a tournament. 1-perfectly orientable graphs form a common generalization of chordal graphs and circular arc graphs. Even though they can be recognized in polynomial time, little is known about their structure. In this paper, we develop several results on 1-perfectly orientable graphs. In particular, we: (i) give a characterization of 1-perfectly orientable graphs in terms of edge clique covers, (ii) identify several graph transformations preserving the class of 1-p…
A $k$-colouring of a graph $G$ is an assignment of at most $k$ colours to the vertices of $G$ so that adjacent vertices are assigned different colours. The reconfiguration graph of the $k$-colourings, $\mathcal{R}_k(G)$, is the graph whose vertices are the $k$-colourings of $G$ and two colourings are joined by an edge in $\mathcal{R}_k(G)$ if they differ in colour on exactly one vertex. For a $k$-colourable graph $G$, we investigate the connectivity and diameter of $\mathcal{R}_{k+1}(G)$. It is known that not all weakly chordal graphs have the property that $\mathcal{R}_{k+1}(G)$ is connected.…
In this paper we introduce the 'simultaneous membership problem', defined for any graph class C characterized in terms of representations, e.g. any class of intersection graphs. Two graphs G_1 and G_2, sharing some vertices X (and the corresponding induced edges), are said to be 'simultaneous members' of graph class C, if there exist representations R_1 and R_2 of G_1 and G_2 that are "consistent" on X. Equivalently (for the classes C that we consider) there exist edges E' between G_1-X and G_2-X such that G_1 \cup G_2 \cup E' belongs to class C. Simultaneous membership problems have applicati…
In this paper, we study the computational complexity of \textsc{$s$-Club Cluster Vertex Deletion}. Given a graph, \textsc{$s$-Club Cluster Vertex Deletion ($s$-CVD)} aims to delete the minimum number of vertices from the graph so that each connected component of the resulting graph has a diameter at most $s$. When $s=1$, the corresponding problem is popularly known as \sloppy \textsc{Cluster Vertex Deletion (CVD)}. We provide a faster algorithm for \textsc{$s$-CVD} on \emph{interval graphs}. For each $s\geq 1$, we give an $O(n(n+m))$-time algorithm for \textsc{$s$-CVD} on interval graphs with …
The problem of computing induced subgraphs that satisfy some specified restrictions arises in various applications of graph algorithms and has been well studied. In this paper, we consider the following Balanced Connected Subgraph (shortly, BCS) problem. The input is a graph $G=(V,E)$, with each vertex in the set $V$ having an assigned color, "red" or "blue". We seek a maximum-cardinality subset $V'\subseteq V$ of vertices that is color-balanced (having exactly $|V'|/2$ red nodes and $|V'|/2$ blue nodes), such that the subgraph induced by the vertex set $V'$ in $G$ is connected. We show that t…
Let $G$ be a connected graph. The eccentricity of a path $P$, denoted by ecc$_G(P)$, is the maximum distance from $P$ to any vertex in $G$. In the \textsc{Central path} (CP) problem our aim is to find a path of minimum eccentricity. This problem was introduced by Cockayne et al., in 1981, in the study of different centrality measures on graphs. They showed that CP can be solved in linear time in trees, but it is known to be NP-hard in many classes of graphs such as chordal bipartite graphs, planar 3-connected graphs, split graphs, etc. We investigate the path eccentricity of a connected graph~…
Let $G(V,E)$ be a simple, undirected and connected graph. A dominating set $S \subseteq V(G)$ is called a $2$-\textit{secure dominating set} ($2$-SDS) in $G$, if for every pair of distinct vertices $u_1,u_2 \in V(G)$ there exists a pair of distinct vertices $v_1,v_2 \in S$ such that $v_1 \in N[u_1]$, $v_2 \in N[u_2]$ and $(S \setminus \{v_1,v_2\}) \cup \{u_1,u_2 \}$ is a dominating set in $G$. The $2$\textit{-secure domination number} denoted by $γ_{2s}(G)$, equals the minimum cardinality of a $2$-SDS in $G$. Given a graph $ G$ and a positive integer $ k,$ the $ 2 $-Secure Domination ($ 2 $-SD…
For a graph $G=(V,E)$, a subset $D$ of vertex set $V$, is a dominating set of $G$ if every vertex not in $D$ is adjacent to atleast one vertex of $D$. A dominating set $D$ of a graph $G$ with no isolated vertices is called a paired dominating set (PD-set), if $G[D]$, the subgraph induced by $D$ in $G$ has a perfect matching. The Min-PD problem requires to compute a PD-set of minimum cardinality. The decision version of the Min-PD problem remains NP-complete even when $G$ belongs to restricted graph classes such as bipartite graphs, chordal graphs etc. On the positive side, the problem is effic…
We consider the problem of devising algorithms to count exactly the number of independent sets of a graph G . We show that there is a polynomial time algorithm for this problem when G is restricted to the class of strongly orderable graphs, a superclass of chordal bipartite graphs. We also show that such an algorithm exists for graphs of bounded clique-width. Our results extends to a more general setting of counting independent sets in a weighted graph and can be used to count the number of independent sets of any given size k .
The partition of graphs into "nice" subgraphs is a central algorithmic problem with strong ties to matching theory. We study the partitioning of undirected graphs into same-size stars, a problem known to be NP-complete even for the case of stars on three vertices. We perform a thorough computational complexity study of the problem on subclasses of perfect graphs and identify several polynomial-time solvable cases, for example, on interval graphs and bipartite permutation graphs, and also NP-complete cases, for example, on grid graphs and chordal graphs.
In a graph, a perfect matching cut is an edge cut that is a perfect matching. Perfect Matching Cut (PMC) is the problem of deciding whether a given graph has a perfect matching cut, and is known to be NP-complete. We revisit the problem and show that PMC remains NP-complete when restricted to bipartite graphs of maximum degree 3 and arbitrarily large girth. Complementing this hardness result, we give two graph classes in which PMC is polynomial time solvable. The first one includes claw-free graphs and graphs without an induced path on five vertices, the second one properly contains all chorda…
We consider the following problem: for a given graph G and two integers k and d, can we apply a fixed graph operation at most k times in order to reduce a given graph parameter $π$ by at least d? We show that this problem is NP-hard when the parameter is the independence number and the graph operation is vertex deletion or edge contraction, even for fixed d=1 and when restricted to chordal graphs. We also give a polynomial time algorithm for bipartite graphs when the operation is edge contraction, the parameter is the independence number and d is fixed. Further, we complete the complexity dich…
