UNITED STATES
SECURITIES AND EXCHANGE COMMISSION
Washington, D.C.  20549

SCHEDULE 13D
Under the Securities Exchange Act of 1934

(Amendment No. 76)*


Valhi, Inc.
(Name of Issuer)

Common Stock, par value $0.01 per share
(Title of Class of Securities)

918905 10 0
(CUSIP Number)

Robert D. Graham
Three Lincoln Centre
Suite 1700
5430 LBJ Freeway
Dallas, Texas   75240-2694
(972) 233-1700
(Name, Address and Telephone Number of Person
Authorized to Receive Notices and Communications)

May 7, 2018
(Date of Event which requires Filing
of this Statement)

If the filing person has previously filed a statement on Schedule 13G to report the acquisition that is the subject of this Schedule 13D, and is filing this schedule because of sections 240.13d-1(e), 240.13d-1(f) or 240.13d-1(g), check the following box. ☐

*The remainder of this cover page shall be filled out for a reporting person's initial filing on this form with respect to the subject class of securities, and for any subsequent amendment containing information which would alter disclosures provided in a prior cover page.

The information required on the remainder of this cover page shall not be deemed to be "filed" for the purpose of Section 18 of the Securities Exchange Act of 1934 ("Act") or otherwise subject to the liabilities of that section of the Act but shall be subject to all other provisions of the Act (however, see the Notes).

(Continued on following pages)
CUSIP No. 918905 10 0

1
NAMES OF REPORTING PERSONS
Valhi Holding Company
2
CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP (SEE INSTRUCTIONS)
(a)  ☐
(b)  ☐
3
SEC USE ONLY
 
4
SOURCE OF FUNDS (SEE INSTRUCTIONS)
OO
5
CHECK IF DISCLOSURE OF LEGAL PROCEEDINGS IS REQUIRED
PURSUANT TO ITEMS 2(d) OR 2(e)  ☐
 
6
CITIZENSHIP OR PLACE OF ORGANIZATION
Delaware
NUMBER OF
SHARES
BENEFICIALLY
OWNED BY
EACH
REPORTING
PERSON
WITH
7
SOLE VOTING POWER
                -0-
8
SHARED VOTING POWER
310,346,282
9
SOLE DISPOSITIVE POWER
                -0-
10
SHARED DISPOSITIVE POWER
310,346,282
11
AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON
310,346,282
12
CHECK IF THE AGGREGATE AMOUNT IN ROW (11) EXCLUDES CERTAIN SHARES (SEE INSTRUCTIONS)  ☐
 
13
PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW (11)
91.5%
14
TYPE OF REPORTING PERSON (SEE INSTRUCTIONS)
CO
CUSIP No.  918905 10 0

1
NAMES OF REPORTING PERSONS
Dixie Rice Agricultural L.L.C.
2
CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP (SEE INSTRUCTIONS)
(a)  ☐
(b)  ☐
3
SEC USE ONLY
 
4
SOURCE OF FUNDS (SEE INSTRUCTIONS)
Not applicable
5
CHECK IF DISCLOSURE OF LEGAL PROCEEDINGS IS REQUIRED
PURSUANT TO ITEMS 2(d) OR 2(e)  ☐
 
6
CITIZENSHIP OR PLACE OF ORGANIZATION
Louisiana
NUMBER OF
SHARES
BENEFICIALLY
OWNED BY
EACH
REPORTING
PERSON
WITH
7
SOLE VOTING POWER
                -0-
8
SHARED VOTING POWER
310,346,282
9
SOLE DISPOSITIVE POWER
                -0-
10
SHARED DISPOSITIVE POWER
310,346,282
11
AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON
310,346,282
12
CHECK IF THE AGGREGATE AMOUNT IN ROW (11) EXCLUDES CERTAIN SHARES (SEE INSTRUCTIONS)  ☐
 
13
PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW (11)
91.5%
14
TYPE OF REPORTING PERSON (SEE INSTRUCTIONS)
OO
CUSIP No.  918905 10 0

1
NAMES OF REPORTING PERSONS
Contran Corporation
2
CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP (SEE INSTRUCTIONS)
(a)  ☐
(b)  ☐
3
SEC USE ONLY
 
4
SOURCE OF FUNDS (SEE INSTRUCTIONS)
OO
5
CHECK IF DISCLOSURE OF LEGAL PROCEEDINGS IS REQUIRED
PURSUANT TO ITEMS 2(d) OR 2(e)  ☐
 
6
CITIZENSHIP OR PLACE OF ORGANIZATION
Delaware
NUMBER OF
SHARES
BENEFICIALLY
OWNED BY
EACH
REPORTING
PERSON
WITH
7
SOLE VOTING POWER
                -0-
8
SHARED VOTING POWER
310,346,282
9
SOLE DISPOSITIVE POWER
                -0-
10
SHARED DISPOSITIVE POWER
310,346,282
11
AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON
310,346,282
12
CHECK IF THE AGGREGATE AMOUNT IN ROW (11) EXCLUDES CERTAIN SHARES (SEE INSTRUCTIONS)  ☐
 
13
PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW (11)
91.5%
14
TYPE OF REPORTING PERSON (SEE INSTRUCTIONS)
CO
CUSIP No.  918905 10 0

1
NAMES OF REPORTING PERSONS
Lisa K. Simmons
2
CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP (SEE INSTRUCTIONS)
(a)  ☐
(b)  ☐
3
SEC USE ONLY
 
4
SOURCE OF FUNDS (SEE INSTRUCTIONS)
OO
5
CHECK IF DISCLOSURE OF LEGAL PROCEEDINGS IS REQUIRED
PURSUANT TO ITEMS 2(d) OR 2(e)  ☐
 
6
CITIZENSHIP OR PLACE OF ORGANIZATION
USA
NUMBER OF
SHARES
BENEFICIALLY
OWNED BY
EACH
REPORTING
PERSON
WITH
7
SOLE VOTING POWER
                -0-
8
SHARED VOTING POWER
310,346,282
9
SOLE DISPOSITIVE POWER
                -0-
10
SHARED DISPOSITIVE POWER
310,346,282
11
AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON
-0-
12
CHECK IF THE AGGREGATE AMOUNT IN ROW (11) EXCLUDES CERTAIN SHARES (SEE INSTRUCTIONS)  ý
 
13
PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW (11)
0.0%
14
TYPE OF REPORTING PERSON (SEE INSTRUCTIONS)
IN
CUSIP No.  918905 10 0

1
NAMES OF REPORTING PERSONS
Serena Simmons Connelly
2
CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP (SEE INSTRUCTIONS)
(a)  ☐
(b)  ☐
3
SEC USE ONLY
 
4
SOURCE OF FUNDS (SEE INSTRUCTIONS)
OO
5
CHECK IF DISCLOSURE OF LEGAL PROCEEDINGS IS REQUIRED
PURSUANT TO ITEMS 2(d) OR 2(e)  ☐
 
6
CITIZENSHIP OR PLACE OF ORGANIZATION
USA
NUMBER OF
SHARES
BENEFICIALLY
OWNED BY
EACH
REPORTING
PERSON
WITH
7
SOLE VOTING POWER
          56,136
8
SHARED VOTING POWER
310,346,282
9
SOLE DISPOSITIVE POWER
          56,136
10
SHARED DISPOSITIVE POWER
310,346,282
11
AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON
56,136
12
CHECK IF THE AGGREGATE AMOUNT IN ROW (11) EXCLUDES CERTAIN SHARES (SEE INSTRUCTIONS)  ý
 
13
PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW (11)
0.0%
14
TYPE OF REPORTING PERSON (SEE INSTRUCTIONS)
IN

AMENDMENT NO. 76
TO SCHEDULE 13D

Capitalized terms not otherwise defined in this Amendment No. 76 to this Statement on Schedule 13D (this "Statement") shall have the meanings assigned to such terms in Amendment No. 73 to this Statement.  This Statement relates to the common stock, par value $0.01 per share (the "Shares"), of Valhi, Inc., a Delaware corporation ("Valhi").  Items 2, 4, 5 and 6 of this Statement are hereby amended as set forth below.  The Reporting Persons (as defined below) are filing this amendment as a result of certain transfers of Shares by VHC (as defined below) in payment for redemption of preferred stock of VHC.

Item 2.
Identity and Background.

Item 2 is amended and supplemented as follows.

(a)
The following entities or persons are filing this Statement (collectively, the "Reporting Persons"):

·
Valhi Holding Company ("VHC") as direct holders of Shares;

·
Dixie Rice Agricultural L.L.C. ("Dixie Rice") and Contran Corporation ("Contran") by virtue of their direct or indirect ownership of Valhi;

·
Lisa K. Simmons by virtue of her being a co-trustee of the Family Trust (as defined and described below) and her position as co-chair of the Contran board of directors (the "Contran Board"); and

·
Serena Simmons Connelly by virtue of being co-trustee of the Family Trust and her position as co-chair of the Contran Board.


By signing this Statement, each Reporting Person agrees that this Statement is filed on its or her behalf.

The following is a description of the relationships among the Reporting Persons.

All of Contran's outstanding voting stock is held by a family trust (the "Family Trust") established for the benefit of Ms. Simmons and Ms. Connelly and their children, for which Ms. Simmons and Ms. Connelly are co-trustees, or is held directly by Ms. Simmons and Ms. Connelly or entities related to them.  Ms. Simmons and Ms. Connelly are sisters.

Ms. Simmons and Ms. Connelly also serve as co-chairs of the Contran Board, and one other member of Contran management also serves on the Contran Board.  As co-trustees of the Family Trust, Ms. Simmons and Ms. Connelly has the shared power to vote and direct the disposition of the shares of Contran stock held by the Family Trust, and Ms. Simmons and Ms. Connelly each have the power to vote and direct the disposition of shares of Contran stock they hold directly or which is held by other entities related to them.

Contran is the holder of 100% of the outstanding common stock of Dixie Rice and may be deemed to control Dixie Rice.  Dixie Rice is the direct holder of 100% of the outstanding common stock of VHC and may be deemed to control VHC.

Ms. Simmons and Ms. Connelly (or a trust for which Ms. Connelly is the sole trustee and sole beneficiary, whose Shares are combined with those held directly by Ms. Connelly for purposes of the percentage below) directly hold, or are related to the following persons or entities that directly hold, the following percentages of the 339,170,949 Shares outstanding as of the close of business on May 15, 2018 based on information from Valhi (the "Outstanding Shares"):

VHC 
    91.5%
Serena Simmons Connelly 
Less than 1%

VHC may be deemed to control Valhi.  Contran may be deemed to control VHC, by virtue of its ownership of Dixie Rice shares.

Ms. Simmons and Ms. Connelly directly hold, or are related to the following persons or entities that directly hold, the following percentages of the outstanding shares of Kronos Worldwide common stock:

Valhi 
50.0%
NLKW Holding LLC ("NLKW") 
30.4%
Contran 
Less than 1%
Serena Simmons Connelly 
Less than 1%

Together, Valhi, NL Industries, Inc. ("NL") (and its wholly-owned subsidiary NLKW) and Contran may be deemed to control Kronos Worldwide.

Ms. Simmons and Ms. Connelly directly hold, or are related to the following persons or entities that directly hold, the following percentages of the outstanding shares of NL common stock:

Valhi 
82.9%
Kronos Worldwide 
Less than 1%
Serena Simmons Connelly 
Less than 1%

Together, Valhi and Kronos Worldwide may be deemed to control NL.

NL (including a wholly owned subsidiary of NL) and Kronos Worldwide own 14,372,970 Shares and 1,724,916 Shares, respectively.  As already stated, Valhi is the direct holder of approximately 82.9% of the outstanding shares of common stock of NL and 50.0% of the outstanding shares of Kronos Worldwide common stock.  As a result of Valhi's direct and indirect ownership of NL and Kronos Worldwide and pursuant to Delaware law and Section 13(d)(4) of the Exchange Act, Valhi treats the Shares that NL and Kronos Worldwide own as treasury stock for voting purposes.  For the purposes of this statement, such Shares that NL and Kronos Worldwide hold are not deemed outstanding.

By virtue of the stock ownership of each of Kronos Worldwide, NL, Valhi, VHC, Dixie Rice and Contran, the role of Ms. Simmons and Ms. Connelly as co-trustees of the Family Trust, Ms. Simmons and Ms. Connelly being beneficiaries of the Family Trust, the direct holdings of Contran voting stock by each of Ms. Simmons, Ms. Connelly and entities related to them, the positions as co-chairs of the Contran Board by each of Ms. Simmons and Ms. Connelly, in each case as described above, (a) Ms. Simmons and Ms. Connelly may be deemed to control the Family Trust, Contran, Dixie Rice, VHC, Valhi, NL and Kronos Worldwide and (b) Ms. Simmons and Ms. Connelly, Contran, Dixie Rice, VHC, Valhi, NL and Kronos Worldwide may be deemed to possess indirect beneficial ownership of, and a pecuniary interest in, shares of common stock directly held by such entities, including any Shares.  However, Ms. Simmons and Ms. Connelly each disclaims beneficial ownership of, and such pecuniary interest in, such shares beneficially owned, directly or indirectly, by any of such entities, except to the extent of their direct beneficial ownership in shares of such entities.

The Reporting Persons understand, based on ownership filings with the SEC or upon information provided by the persons listed on Schedule B to this Statement, that such persons may be deemed to own beneficially the Shares as indicated on Schedule C to this Statement.

Item 4.
Purpose of Transaction

Item 4 is amended and supplemented as follows.

Depending upon their evaluation of Valhi's business and prospects, and upon future developments (including, but not limited to, performance of the Shares in the market, availability of funds, alternative uses of funds, and money, stock market and general economic conditions), any of the Reporting Persons or other entities that may be deemed to be related to Contran may from time to time purchase Shares, and any of the Reporting Persons or other entities that may be deemed to be related to Contran may from time to time dispose of all or a portion of the Shares held by such person, or cease buying or selling Shares.  Any such additional purchases or sales of Shares may be in open market or privately negotiated transactions or otherwise.

The information included in Item 2 of this Statement is hereby incorporated herein by reference.  As described under Item 2 of this Statement, Ms. Simmons and Ms. Connelly may be deemed to control Valhi.

Except as described in this Item 4, none of the Reporting Persons nor, to the best knowledge of such persons, any other person named in Schedule B to this Statement has formulated any plans or proposals which relate to or would result in any matter required to be disclosed in response to paragraphs (a) through (j) of Item 4 of Schedule 13D.

Item 5.
Interest in Securities of the Issuer.

Item 5 is amended and supplemented as follows.

(a) VHC and Ms. Connelly (or a trust for which Ms. Connelly is the sole trustee and sole beneficiary, whose Shares are combined with those held directly by Ms. Connelly in the table below) directly hold the following Shares:

Reporting Person
  
Shares Directly Held
  
       
VHC 
    
310,346,282
  
Serena Simmons Connelly 
    
56,136
  
Total 
    
310,402,418
  

By virtue of the relationships described under Item 2 of this Statement:

(1) VHC, Dixie Rice, Contran and Lisa K. Simmons may each be deemed to be the beneficial owner of the 310,346,282 Shares (approximately 91.5% of the Outstanding Shares) that VHC directly holds; and

(2) Serena Simmons Connelly may be deemed to be the beneficial owner of the 310,402,418 Shares (approximately 91.5% of the Outstanding Shares) that VHC and she directly hold, or a trust for which she is sole trustee and sole beneficiary directly holds.

Except for 56,136 Shares (the "Connelly Shares") she holds directly or through a trust for which she is sole trustee and sole beneficiary, Ms. Connelly disclaims beneficial ownership of all Shares.  Ms. Lisa Simmons disclaims beneficial ownership of all Shares.

(b) By virtue of the relationships described in Item 2 of this Statement:

(1) VHC, Dixie Rice, Contran, Ms. Connelly and Ms. Lisa Simmons may each be deemed to share the power to vote and direct the disposition of the 310,346,282 Shares (approximately 91.5% of the Outstanding Shares) that VHC directly holds; and

(2) Ms. Connelly may be deemed to have sole power to vote and direct the disposition of the 56,135 Shares (approximately 0.0% of the Outstanding Shares) that she directly holds, or a trust for which she is sole trustee and sole beneficiary directly holds.

(c) On May 5, 2018, VHC exercised its right to redeem 5,277 shares (the "Redeemed Shares") of its outstanding non-voting 2% Series A Convertible Preferred Stock, par value $1.00 per share (the "Series A Preferred Stock"), at a redemption price of $5,000.00 per share (the "Redemption Price").  Payment of the Redemption Price was made, at the election of VHC and pursuant to the terms of the Certificate of Designation ("Certificate of Designation") for the Series A Preferred Stock, in 3,686,866 Shares (valued at $7.17 for purposes of calculating the number of Shares pursuant to the terms of the Certificate of Designation) and cash for fractional shares.  The 3,686,866 Shares were transferred effective May 7, 2018 as part of payment of the Redemption Price.

(d) VHC and Ms. Connelly each has the right to receive and the power to direct the receipt of dividends from, and proceeds from the sale of, the Shares directly held by such entity or person (or, in the case of Ms. Connelly, by the trust for which she is sole trustee and sole beneficiary).

Item 6.
Contracts, Arrangements, Understandings or Relationships with Respect to Securities of the Issuer.

Item 6 is amended and supplemented as follows.

The information included in Items 2, 4 and 5 of this Statement is hereby incorporated herein by reference.

To the best of their knowledge, none of the Reporting Persons or any person named in Schedule B to this Statement has any contract, arrangement, understanding or relationship (legal or otherwise) with any person with respect to Shares, including, but not limited to, transfer or voting of any such securities, finder's fees, joint ventures, loans or option arrangements, puts or calls, guarantees of profits, division of profits or losses, or the giving or withholding of proxies.
Signature

After reasonable inquiry and to the best of my knowledge and belief, I certify that the information set forth in this Statement is true, complete and correct.

Date: May 17, 2018




/s/ Serena Simmons Connelly 
Serena Simmons Connelly, Signing in the capacities listed on Schedule "A" attached hereto and incorporated herein by reference.

Signature

After reasonable inquiry and to the best of my knowledge and belief, I certify that the information set forth in this Statement is true, complete and correct.

Date: May 17, 2018




/s/ Lisa K. Simmons 
Lisa K. Simmons
Signing in the capacities listed on Schedule "A" attached hereto and incorporated herein by reference.

Signature

After reasonable inquiry and to the best of my knowledge and belief, I certify that the information set forth in this Statement is true, complete and correct.

Date: May 17, 2018




/s/ Robert D. Graham 
Robert D. Graham, Signing in the capacities listed on Schedule "A" attached hereto and incorporated herein by reference.
SCHEDULE A


LISA K. SIMMONS, as CO-TRUSTEE of the Family Trust and as CO-CHAIR of the Contran Board.

SERENA SIMMONS CONNELLY, as CO-TRUSTEE of the Family Trust, as CO-CHAIR of the Contran Board, and as holder (directly or indirectly) of the Connelly Shares.

ROBERT D. GRAHAM, as PRESIDENT and CHIEF EXECUTIVE OFFICER of Contran and CHAIRMAN OF THE BOARD, PRESIDENT and CHIEF EXECUTIVE OFFICER of each of:

DIXIE RICE AGRICULTURAL L.L.C.
VALHI HOLDING COMPANY


SCHEDULE B


The names of the directors and executive officers of the Reporting Persons and their present principal occupations are set forth below.  Each such person is a citizen of the United States of America and the business address of each such person is 5430 LBJ Freeway, Suite 1700, Dallas, Texas  75240.

Name
Present Principal Occupation
   
Serena S. Connelly 
Director and co-chair of the board of Contran.
   
Robert D. Graham 
Vice chairman of the board, president and chief executive officer of Valhi and Kronos; vice chairman of the board and chief executive officer of NL; director and executive vice president of CompX; chairman of the board, president and chief executive officer of Dixie Rice and VHC; and director, president and chief executive officer of Contran.
   
Kelly D. Luttmer 
Executive vice president and chief tax officer of Valhi, NL, Kronos, CompX, Dixie Rice and Contran; and director, executive vice president and chief tax officer of VHC.
   
Andrew B. Nace 
Executive vice president, general counsel and secretary of Valhi, Dixie Rice and Contran; executive vice president of NL, Kronos and CompX; and director, executive vice president, general counsel and secretary of VHC.
   
Lisa K. Simmons 
Director and co-chair of the board of Contran.
   
Gregory M. Swalwell 
Executive vice president, chief financial officer and chief accounting officer of Valhi, Dixie Rice and Contran; executive vice president and chief financial officer of NL and Kronos; executive vice president of CompX; and director, executive vice president, chief financial officer and chief accounting officer of VHC.
SCHEDULE C

Based upon ownership filings with the Commission or upon information provided by the persons listed on Schedule B to this Statement, such persons may be deemed to personally beneficially own Shares, as outlined below:

Name
Shares Held
   
Serena S. Connelly 
56,136
(1)
     
Robert D. Graham 
4,000
  
     
Kelly D. Luttmer 
-0-
  
     
Andrew B. Nace 
-0-
  
     
Lisa K. Simmons 
-0-
(2)
     
Gregory M. Swalwell 
3,498
  


(1)
Does not include other Shares of which Ms. Connelly may be deemed to possess indirect beneficial ownership as described in Items 2 and 5(a) of this Statement.  Except for the 56,136 Shares she holds directly or through a trust for which she is sole trustee and sole beneficiary, Ms. Connelly disclaims beneficial ownership of all Shares.

(2)
Does not include Shares of which Ms. Lisa Simmons may be deemed to possess indirect beneficial ownership as described in Items 2 and 5(a) of this Statement.  Ms. Lisa Simmons disclaims beneficial ownership of all Shares.